Quantum Invariants of Calabi-Yau threefolds · space of the vector bundle is a Calabi-Yau of...

J I M B R YA N

Q U A N T U M I N VA R I A N T SO F C A L A B I - YA U T H R E E -F O L D S

L E C T U R E N O T E S B Y P O OYA R O N A G H

Copyright © 2014 Jim Bryan

lecture notes by pooya ronagh

[email protected]

The following are lecture notes from a course offered at the University of British Columbia (UBC), in spring2014 by Prof. J. Bryan. The script at hand is compiled from lecture notes I took in the class, and are typesetin Tufte book style, tufte-latex.googlecode.com.

First printing, April 2014

Contents

1 Gromov-Witten Theory 13

2 Donaldson-Thomas Theory 33

3 Pandharipande-Thomas Theory 49

4 Gupakumar-Vafa Invariants 55

5 Generalized Donaldson-Thomas Invariants 59

A Solution to some problems 95

List of Figures

1.1 Embedding of the nodal curve is contributes to a unique map 14

1.2 Stable mappings to double line 14

1.3 Nodal genus 1 curve 14

1.4 Some stable maps to the singularity x2 = y315

1.5 Some possible maps with genus one 15

1.6 More possible maps with genus one 15

1.7 Even more possible maps with arithmetic genus one 15

1.8 Characterization of stable maps 16

1.9 Elements of same fiber 17

1.10 Homologous pointed stable maps 20

1.11 Lattice picture 31

1.12 The map contributing to disconnected invariants 31

2.1 Analogue of degenerate maps 33

2.2 Analogue of collapsing components are embedded points 33

2.3 Degeneration of the cubic (x, y) 7→ (x3, x2y, xy2, y3) in P3 to re-duced plane cubic curve C with a node 34

2.4 The local model of the degeneration 34

2.5 Possible deformations in families of curves. The limit can be asmooth curve with an isolated point. 34

2.6 Milnor fiber 37

2.7 Lines in P239

2.8 Conics in P239

2.9 Configurations that asymptotically have one box along the z-axis 40

2.10 Atiyah-Bott localization 40

2.11 The local P140

2.12 Configurations contributing to I2(X, `) 40

2.13 Configurations contributing to I3(X, `) 40

2.14 Interpretation of a 2D-partition as a quotient ring. 45

2.15 The asymptote of λ has contains three cubes in z-direction. 46

6

3.1 Are there limits that do not create embedded points? 49

3.2 Points (given by divisor D) are now constrained to lie on the curve(C) 51

4.1 Curve-counting invariants and their correspondences 55

5.1 Resolution of singularities 74

5.2 Resolution of the singularity ∑ni=1 x2

i = 0 75

5.3 Homogenous map of degree d 78

5.4 Blow up of the cone (the middle fibration) constructs X0 over thecentral fiber X0; the exceptional divisor of X0 is the base varietyof this fibration, { fd = 0} ⊆ Pn−1 in the lower middle picture.This is follow by the embedded resolution of singularities, E0,consisting of normal crossing divisors F1, · · · , Fn in lower righthand picture; The pull back (upper right hand picture) createsE1, · · · , Ek and E0 ⊆ Pn−1 on the base. 78

Introduction

This course is about the quantum invariants themselves rather thanthe geometry of the underlying spaces. In other words, we are notnecessarily going to learn more about the Calabi-Yau threefolds butthe goal is really focusing on the very invariants defined for them. Wewant to:

1. Define these invariants;

2. Study their structure (it turns out that these invariants have veryinteresting internal structures);

3. Study the relationship between different kinds of invariants; and,

4. Study their connections with other branches of mathematics (one ofthe beauties of these invariants is that they turn out to be tied tosome surprising other topics such as modular forms and represen-tations of algebraic groups).

That said, the first step is to define the Calabi-Yau manifolds:Definition 0.1. A Calabi-Yau manifold of dimension n is a complexprojective manifold1 X ⊆ CPN (a nonsingular projective variety) with 1 We are always working over the field

of complex numbers in this course.dimC X = n satisfying any of the following equivalent conditions:1. X admits a Kähler metric whose Ricci curvature is zero (this is called

a Ricci-flat or Calabi-Yau metric)2 2 This is a purely geometric condition.

2. X has a non-vanishing holomorphic n-form;3. The canonical line bundle of X, KX =

∧n T∗X is trivial: KX ∼= OX .Remark 0.2. (1) implying (2) and equivalence of (2) and (3) is easy. (3)implying (1) is Yau’s field medal winner theorem.Remark 0.3. As algebraic geometers we often the (3) as the definitionof a Calabi-Yau, even in the case where X is not compact (so when X isquasi-projective). In this non-compact case (3) does not imply (1) butwe may still consider X to be Calabi-Yau whenever KX ∼= OX .

8

Remark 0.4. Some people might require that H1(X, OX) = 0 maybeexcluding the case of dim X = 1. (1) does imply H1(X, OX) = 0.Example 0.5. In dimX = 1 then Calabi-Yau manifolds are ellipticcurves, i.e. Riemann surfaces of genus 1 which all turn out to be C/Z2

quotients, and i.e. cubic plane curves X(3) ⊆ CP2 and H1(X, OX) ∼= C.There is only one topological type for these (that of a topological torus)but they have a moduli as a complex manifolds.Example 0.6. In dimX = 2, Calabi-Yau manifolds are either Abeliansurfaces (described as C2/Z4, topologically a T(4) = (S1)4) or a K3-surface (e.g. a hyper surface of degree 4 in 3-projective plane: X(4) ⊆CP3. Abelian surfaces have H1(X, OX) 6= 0 but K3 surfaces haveH1(X, OX) = 0. There are two topological types, each type has a mod-uli.Example 0.7. There is a vast number of distinct topological types ofCY3s (possibly infinite, conjectured to be finite). About 5 hundredthousand already known!! A quintic threefold X(5) ⊆ CP4 in an ex-ample of a Calabi-Yau threefold.Example 0.8. More generally, any hypersurface, Xn+1 ⊆ CPn, of de-gree n + 1 is a Calabi-Yau of dimension n− 1.

X(n+1) = {(x0, · · · , xn) ∈ Pn : F(x0, · · · , xn) = 0}

where F is a homogeneous polynomial of degree n + 1.Example 0.9. Generalizing further, if s is a transverse section of thedual canonical bundle K∗M =

∧d−M TM of a smooth projective man-ifold M then s−1(0) is a Calabi-Yau. (This generalizes last exampleif M = CPn and consequently K∗M ∼= O(n + 1), hence sections beinggiven by polynomials of degree n + 1.)Example 0.10. For a construction of non-compact examples, let M bea projective manifold of dimension d and E → M be a holomorphicvector bundle of rank r with ∧nE ∼= KM, then let X = tot(E) totalspace of the vector bundle is a Calabi-Yau of dimension r + d.

0.1 Quantum invariants of Calabi-Yau threefolds

The quantum invariants of Calabi-Yau threefolds are deformation in-variants which have analogous quantities in string theory/QFT. Specific of Calabi-Yau threefolds

is that the expected dimension ofthe spaceM(X, β) of curves inclass β is zero.

A deformation invariant is a quantity (usually a number) associatedto X which is invariant under deformations of the complex structureof X.

The invariants we study arise form counting (in the ideal setting)holomorphic curves C ⊂ X. A curve C ⊂ X gives rise to a homologyclass [C] ∈ H2(X; Z) and in particular we are interested in countingcurves C ⊂ X in a fixed homology class β ∈ H2(X; Z). The key point

9

here is that

The expected dimension of the spaceM(X, β) of curvesC ⊂ X with [C] = β is zero. This is unique to Calabi-Yauthreefolds.

Jan 8

Last time we defined Calabi-Yau manifolds. We will define quan-tum invariants of Calabi-Yau threefolds now. These will be deforma-tion invariants related to quantities in string theory. Our invariantswill arise from counting homolorphic curves C ⊆ X. Typically we fixβ ∈ H2(X; Z) and count curves C ⊆ X such that [C] = β.

To perform this count we will want to construct a moduli spaceM(X, β) parametrizing curves in the class β. In other words eachpoint in M(X, β) corresponds to a curves and in ideal situation thereis a finite number of curves and henceM(X, β) is a zero dimensionalprojective manifold.

As we will see this ideal situation almost never holds. But we willsee how we can get close to having this situation being the case. IfX is a Calabi-Yau threefold, then the expected dimension of M(X, β) isalways zero (regardless of X and β). This is one of the reasons whyCalabi-Yau threefolds are special.

0.2 Expected dimension

If a variety M is given by the zero locus of a set of equations on someambient smooth space V, then the expected dimension in this situationis dim V− #{ equation }. If the solutions to the equations intersectionstransversally then M is a smooth manifold of dimension equal to theabove number. But otherwise (i.e. when M has excess intersection),M may be singular, and it may be larger than expected dimension. Sovaguely speaking, the main issues to overcome are:1. Deal with the excess dimension of moduli space.2. We will need compactness or projectivity (i.e. a compactification).Different invariants arise from different ways of dealing with these is-sues.

Blue print of a curve-counting theory

Xhanding issue (1)

//M(X, β)handling issue (2)

// invariants

Calabi-Yauthreefold

Some compact modulispace of curves [C] ⊆ Xwith [C] = β

Usually the degree of a vir-tual class, and sometimesdimension of a cohomol-ogy group of this space

10

There are two basic strategies:1. View C ⊂ X as being parametrized f : C → X and keep C as non-

singular as possible, but let the map no longer be an embedding.This leads to the moduli space of stable maps in Gromov-Witten the-ory. In physics, these correspond to world sheets of strings (wherea one-dimension string travels in time to sweep out a two-dimensionRiemannian surface called a world sheet).

2. View C ⊆ X as being cut out be equations. Hence the curves aredescribe in therm of sheaves, giving rise to various moduli spacesof sheave.

(a) Moduli space of ideal sheaves: these lead to Donaldson-Thomasinvariants.

(b) Moduli space of stable pairs: these lead to Pandharipande-Thomastheory.

(c) Moduli space of torsion sheaves: these lead to Gupakumar-Vafainvariants.

Sheaves (pairs, etc.) in physic are configurations of D-branes (Theseare boundary conditions in string theory).

0.3 Syllabus

The invariants Gromov-Witten (GW), Donaldson-Thomas (DT), Pandharipande-Thomas (PT), and Gupakumar-Vafa (GV) are the main topics of thecourse. We want to talk about1. Their definitions;2. Their structures;3. And the relationship between them.

The outline of the course is hence as follows:1. Invariants from curve counting:

(a) GW invariants (2 weeks)(b) DT invariants (2 weeks)(c) GW/DT correspondence (2 weeks)(d) PT theory and PT/DT correspondence (2 weeks)(e) GV theory and GV conjecture (1 week)

2. Computational methods of the invariants(a) Localization and topological vertex(b) Degeneration and relative invariants (?)(c) Motivic methods

3. Local geometries(a) Local curves (this leads to the interesting TQFT)(b) Local surfaces and Göttsche conjectures

4. Orbifold Calabi-Yau threefolds and their resolutions5. Refined invariants and wall-crossing techniques.

11

0.4 Course references

A nice overview on curve-counting invariants is written by Pandhari-pande and Thomas 3. For Gromov-Witten invariants many sources 3 R. Pandharipande and R. P. Thomas.

13/2 ways of counting curves. ArXive-prints, November 2011

such as the relevant chapters of Mirror Symmetry of Clay 4 is a read-4 Kentaro Hori, Sheldon Katz, AlbrechtKlemm, Rahul Pandharipande, RichardThomas, Cumrun Vafa, Ravi Vakil, andEric Zaslow. Mirror symmetry, volume 1

of Clay Mathematics Monographs. Amer-ican Mathematical Society, Providence,RI; Clay Mathematics Institute, Cam-bridge, MA, 2003. With a preface byVafa

able introduction. For Donaldson-Thomas invariants I would refer youto 5. For Pandharipande-Thomas invariants the papers 6, 7 and 8 are

5 D. Maulik, N. Nekrasov, A. Okounkov,and R. Pandharipande. Gromov-Wittentheory and Donaldson-Thomas theory,I. ArXiv Mathematics e-prints, December2003

6 R. Pandharipande and R. P. Thomas.Curve counting via stable pairs inthe derived category. InventionesMathematicae, 178:407–447, May 2009

7 R. Pandharipande and R. P. Thomas.The 3-fold vertex via stable pairs. ArXive-prints, September 2007

best references.

Chapter 1Gromov-Witten Theory

Jan 15

In a family of curves in a projective manifold X, a smooth curve candegenerate to a singular curve. For example the smooth conic xy +

z2 = 0 in CP2, we have xy = 0 which are a pair of lines P1 ∪pt P1

which topologically looks like the wedge of two spheres. This is calleda nodal singularity. Another deformation of this curves is z2 = 0 whichis a doubled line.

A smooth cubic in P2 is topologically a torus which can deform toa nodal cubic (i.e. locally xy = 0 around the singularity) which lookstopologically like a P1 with two points of it glued together. The affineequation x2 = y3 in P2 is another type of singularity and there existsa degree one map from a smooth genus 0 curve to it.

Different moduli spaces handle these degenerations differently andlead to very different curve counting theories. In Gromov-Witten the-ory we consider curves as being given by maps f : C → X. We requirethat the domain curve has at worst nodal singularities but we allowthe map to not necessarily be an embedding.

1.1 Moduli of stable maps

Definition 1.1. A stable map to X of genus g and class β ∈ H2(X, Z)

is a map f : C → X where C is a curve of arithmetic genus g with atworst nodal singularities, f∗[C] = β ∈ H2(X, Z) and such that

Aut( f ) := {ϕ : C∼=−→ C, f ◦ ϕ = f }

is finite.Definition 1.2. Two stable maps f : C → X and f ′ : C′ → X are

equivalent if there exists ϕ : C∼=−→ C′ such that the following diagram

14 quantum invariants of calabi-yau threefolds

commutes

C

ϕ��

f// X

C′f ′

??

The reason for this definition is really the following theorem.Theorem 1.3. The moduli space of stable maps of fixed genus and degreedenotedMg(X, β) is a projective Deligne-Mumford stack.

In particular, this means that there is projective variety whose pointsare in bijective correspondence with isomorphism classes of stablemaps.Example 1.4. ConsiderM0(P

2, 2H) where H is the class of a line P1 ⊆P2. There is a map M0(P

2, 2H) → P5 illustrating the fact that theconics in P2 form a linear system of dimension 5.

The nodal curve will be associated to an embedding of P1 ∪pt P1 tothe nodal curve. And this is unique up to isomorphism as shown infigure 1.1.

Figure 1.1: Embedding of the nodalcurve is contributes to a unique map

The fiber of map over a double line 2H ⊆ P5 is a copy of Sym2 P1 =

P1 × P1/Z/2 ∼= P2. When the two branch points come close to jointogether we get a map P1 ∪pt P1 → P1 (figure 1.2).

The locus of double lines in the linear system of conics is hence acopy of P2 sitting in P5 is the Veronese embedding P2 ↪→ P5.

Figure 1.2: Stable map-pings to double line

SoM0(P2, 2[H]) ∼= BlP2⊆P5P5 and the exceptional divisor here are

points with Z/2 automorphisms.

Figure 1.3: Nodal genus 1 curve

gromov-witten theory 15

Example 1.5. What happens in M1(P2, 3[H])? We have embeddings

of genus 1 curves which can deform to nodal ones as well (figure 1.3).1.7.

Figure 1.4: Some stable mapsto the singularity x2 = y3

Figure 1.5: Some possi-ble maps with genus one

The limit of these are singularities locally looking like x2 = y3 (fig-ure 1.4). But even more interesting phenomena can happen as shownin figures 1.5, 1.6 and

Figure 1.6: More possi-ble maps with genus one

Figure 1.7: Even more possiblemaps with arithmetic genus one

Jan 17

Theorem 1.6. The virtual (or expected) dimension ofMg(X, β) is

vdim(Mg(X, β)) = −KX .β + (dimC X− 3)(1− g).

Here note as well that −KX .β = c1(TX)[β].Example 1.7. If dim X = 3 then the virtual dimension is independentof the generous. If X is a Calabi-Yau threefold then the virtual dimen-sion is zero regardless of genus and homology class β.Example 1.8. In P2 we have

vdim(Mg(P2, d[H])) = −O(−3).d[H] + g− 1 = 3d + g− 1.

A short exercise is to show that this formula matches naive count com-ing from dimension of linear system |O(d)| and genus formula.

Suppose that C ⊂ X is a smooth curve of genus g and degree β withnormal bundle NC/X → X. A deformation of C can be thought of as asection of NC/X . The long exact sequence associated to

0→ TC → TX |C → NC/X → 0

where TX |C = f ∗TX , is

0→ H0(C, TC)→ H0(C, TX)→ H0(C, N)

→ H1(C, TC)→ H1(C, TX)→ H1(C, N)→ 0.

Here H0(C, N) = Def( f : C → X) and H1(C, TC) = Def(C) is the spaceof deformations of C. The group H1(C, TX) is the space of obstructionsOb( f ) and H1(C, N) = Ob( f : C → X). Also H0(C, TC) = Aut(C) andH0(C, TX) = Def( f ). So the actual dimension of Tf :C→XMg(X, β) =

dim H0(C, N). And the expected dimension is

h0(C, N)− h1(C, N) = deg(N) + dim N(1− g)

= −KX .β + 2g− 2g + (dim X− 1)(1− g)

= −KX .β + (dim X− 3)(1− g)

This calculation is valid when C is smooth. But of course in generalC can be nodal and f : C → X does not have to be an embeddingeither.


Remark 1.9. If M is a variety, the Zariski tangent space Tp M of M atp can be given by the set of maps Spec(C[x]/(x2)) → M that sendthe closed point of the domain to p. Hence T[ f :C→X]M is the spaceof families of maps over Spec C[x]/(x2). Calculating this space oftenturns out to have a homological method and is the topic of discussionin deformation theory.

A way to view this dimension formula is by looking at the modulispace of all nodal curves Mg which is smooth and of dimension 3g− 3by Riemann-Roch and look at the fibbers of the map

p :Mg(X, β)→ Mg

and observe that

vdim = dim Mg + vdim(fiber of p at [C]).

The latter is again by Riemann-Roch just

h0(c, f ∗TX)− h0(C, f ∗TX) = c1( f ∗TX).[C] + dim X.(1− g)

= c1(TX). fX [C] + dim X(1− g)

= −KX .β + dim X(1− g).

Mg(pt, 0) = Mg = {C : possibly nodal of genus g , |Aut(C)| < ∞}.

This is a smooth compact orbital of dimension 3g− 3. The mapMg(X, β)→Mg sending [ f : C → X] to Cstab where we take C and contract com-ponents with automorphisms.

Concretely what does |Aut( f : C → X)| < ∞ mean? It means thatf : C → X is stable if every collapsing components Ci of genus zerohas at least 3 nodes (and M1(X, 0) is empty!).

Figure 1.8: Character-ization of stable maps

So for instance for M2(P2, [H]) → M2 we have the situation de-

picted in figure 1.9.Jan 20

1.2 Marked points

We define a new type of moduli spaceMg,n(X, β) via

Mg,n(X, β) =

f : (C, x1, · · · , xn)→ X,(possibly nodal) genus g curvewith n non-singular distinct points

x1, · · · , xn ∈ C and |Aut( f )| < ∞

.

(1.1)

Stability then is the condition of every genus zero collapsing compo-nent has 3 or more points which are nodes or marked (and by defini-tionM1,0(X, 0) = ∅). As a notationMg,n :=Mg,n(pt, 0).


Figure 1.9: Elements of same fiber

Deligne-Mumford moduli space of stable curves are smooth orb-ifolds of dimension 3g− 3 + n, and are define whenever 2g + n ≥ 3.Example 1.10. M0,n is a manifold of dimension n− 3.

M0,n =

:

trees of P1’s with3 marked or nodalpoints on eachcomponent

.

Example 1.11. M0,n is a manifold of dimension n− 3. With only threemarked points the moduli space is a single point containing the pro-jective line with three marked points on it:

M0,3 =

In case of 4 marked points we have a strata of P1 r {0, 1, ∞} and threepoints in the limits

M0,4 =

⋃

⋃

⋃

.


which turns out to beM0,4 = P1.We have a forgetful map

Mg,n(X, β)→Mg,n

mapping f : (C, x1, · · · , xn) → X to (C, x1, · · · , xn)stab i.e. the curvewith marked points as before except with non-stable components hav-ing been contracted. In particular relations in the cohomology ofMg,n

pullback to relations on Mg,n(X, β). The results are (universal) rela-tions among Gromov-Witten invariants.

1.3 Definition of Gromov-Witten invariants

The moduli spaceMg,n(X, β) also has maps

evi :Mg,n(X, β)→ X

given by mapping f : (C, x1, · · · , xn)→ X to f (xi).Suppose for instance that we want to count the number of curves

passing through two points p and q:

ev−11 (p) ∩ ev−1

2 (q) ⊆M0,2(P2, H)

is the intersection of loci of maps where x1 maps to p and x2 maps toq. More generally if A1, · · · , An ⊂ X are fixed submanifolds and

ev−11 (A1) ∩ · · · ∩ ev−1

n (An) ⊆Mg,n(X, β)

is finite then the cardinality of this set is the number of genus g curvesof degree β passing through A1, · · · , An.

Dually we define

NGWg,β (A1, · · · , An) =

∫[Mg,n(X,β)]

ev∗1(PD(A1)) ∪ · · · ∪ ev∗n(PD(An)).

(1.2)

But the fundamental class [Mg,n(X, β)] and integration exist ifMg,n(X, β)

was smooth. In this case the above is an element of

H2 dimMg,n(X,β)(Mg,n(X, β), Z).

To make definition 1.2 work in general we need something thatplays the role of the fundamental class in the smooth case.Theorem 1.12. There exists a class [Mg,n(X, β)]vir ∈ H∗(X, Q) the virtualfundamental class of degree 2 vdimMg,n(X, β).

Suppose we are given E→ Y a vector bundle of rank r on a projec-tive manifold of dimension d. We define a variety M by M = s−1(0) ⊂Y. The M has the expected dimension d − r if s is transverse to the


zero section. In this case M is a manifold of dimension d − r. If thefundamental class [M] satisfies

i∗[M] = PD(cr(E)︸︷︷︸∈H2r

)

︸︷︷︸∈H2d−2r

∈ H2(d−r)(Y, Z).

If s is not transverse, then M can be singular and/or larger than ex-pected dimension but there is a class [M]vir ∈ H2(d−r)(M) and satisfy-ing i∗[M]vir = PD(cr(E)).

Note that [Mg,n(X, β)]vir has Q coefficients because the fundamen-tal class of an orbifold is only define over Q: so NGW

g,β (A1, · · · , An) ∈ Q.

The number NGWg,β (A1, · · · , An) is zero unless

−KX .β + (dim X− 3)(1− g) + n = codimC A1 + · · ·+ codimC An.

Each codim Ai cycle in X imposes codim Ai − 1 conditions on curvesin X.Example 1.13. If codim A = 1, meaning A is a divisor, then askingthat a curves passes through A imposes no conditions. (The numberof intersections is determined cohomologically.) Asking a curve topass through a point on a surface imposes one condition.Example 1.14. If X is a Calabi-Yau threefold, the we need no conditions

NGWg,β :=

∫[Mg,0(X,β)]vir

1 ∈ Q.

Gromov-Witten invariants are the closest to their enumerative inter-pretation when Mg,n(X, β) is smooth and of the expected dimension.Best set of examples is g = 0 and X = Pn. In this case H1(C, f ∗TPn) =

0 for all ( f : C → X) ∈ M0,0(Pn, d[`]). In other words, the obstructions

vanish!If X = P2 then dimM0,0(P

2, d[`]) = 3d− 1

Nd = NGW0,d[`](

3d−1︷︸︸︷pt, · · · , pt) =

number of rational curves of degree dpassing through 3d− 1 points.

Theorem 1.15 (Kontsevich). For d > 1 we have

Nd = ∑d1+d2=dd1,d2>0

Nd1 Nd2

(d2

1d22

(3d− 43d1 − 2

)− d3

1d2

(3d− 43d1 − 1

))

He proved this using Quantum Cohomology of P2. This structure isa deformation of the cup product on H∗(P2) built from genus zeroGromov-Witten invariants.

The above formula follows from the associativity of the quantumproduct, which come from the relations on M0,4 that the objects of


figure 1.3 are homologous. We will discuss this in more details thisweek.

Figure 1.10: Homolo-gous pointed stable maps

Jan 22

1.4 Quantum Cohomology

This section is a digression of the mainstream of the course. It is not somuch relevant to Calabi-Yau threefolds but interesting to get familiarwith. As we said last time Quantum Cohomology is a deformationof the cup product on H∗(X) depending on parameters built fromgenus zero Gromov-Witten invariants of X. The quantum product willbe associative. Associativity arises from a non-trivial relation amongthe invariants. It is inherited from the trivial relationM0,4

∼= P1.First we put all invariants together in a generating function called

Gromov-Witten potential. As a piece of notation we set

〈α1, · · · , αn〉X0,β = NGW0,β (α1, · · · , αn)

for all β ∈ H2(X) and αi ∈ H∗(X).For convenience we assume that the cohomology of X is concen-

trated in even degrees: H∗(X) = Heven(X). Then 〈·, · · · , ·〉X0,β will bemulti-linear and symmetric. So we may use monomial notation.Example 1.16. The number of rational curves of degree d in P2 passingthrough 3d− 1 points, Nd, is given via

Nd = 〈3d−1︷︸︸︷

pt, · · · , pt〉P2

0,d = 〈pt3d−1〉P2

0,d.

Let1︷︸︸︷

T0 ,

H2(X)︷︸︸︷T1, · · · , Tp, · · · , Tm ∈ H∗(X, Z) be a basis. Assume we have

chosen T1, · · · , Tp so that βi = Ti.β ≥ 0 for all curve classes β. Let

gij :=∫

XTi ∪ Tj

and(

gij) be the inverse matrix. Let t0, · · · , tm be formal variables andlet γ denote the summation γ = t0T0 + · · ·+ tmTm.Definition 1.17. The genus zero Gromov-Witten potential is definedvia

F := ∑β

〈exp(γ)〉X0,β qβ where qβ := qβ11 · · · q

βpp

= ∑β

∞

∑n=0

1n!〈(t0T0 + · · ·+ tmTm)

n〉X0,βqβ

= ∑β

∑n0,··· ,nm≥0

〈Tn00 , · · · , Tnm

m 〉X0,βtn00

n0!· · · tnm

mnm!

qβ


which is an element of Q[[q1, · · · , qp, t0, · · · , tm]] as a formal generatingfunction.

Coefficients of the GW potential are all possible genus-zero Gromov-Witten invariants of X. In this generating function ti’s are trackinginsertions and qi’s are tracking β.Note 1.18. The partial derivatives of F are interesting:

∂`F∂ti1 · · · ∂ti`

= ∑β

〈exp(γ)Ti1 · · · Ti`〉0,βqβ

∂`F∂ti1 · · · ∂ti`

∣∣∣∣∣t=0

= ∑β

〈Ti1 · · · Ti`〉0,βqβ

Question: What happens at q = 0, i.e. what are 〈Tn00 · · · T

nmm 〉X0,0?

〈γ1, · · · , γn〉0,0 =∫[M0,n(X,0)]vir ]

ev∗1(γ1) ∪ · · · ∪ ev∗n(γn)

But note thatM0,n(X, 0) = M0,n × X with the mapping

( f : (C, x1, · · · , xn)→ X) 7→ ((C, x1, · · · , xn), im( f )).

and has dimension −KX .β + (dim X − 3)(1− g) + n = n− 3 + dim Xsince g = 0 and β = 0. Hence this space is smooth of expected dimen-sion so the virtual class is the fundamental class of the variety.

In this case evi : M0,n → X is just the projection πX , on the secondfactor. And we have,

〈γ1, · · · , γn〉X0,0 =∫

M0,n×Xπ∗X(γ1) · · ·π∗X(γn)︸︷︷︸

3−n+dim X

=

{0 if n 6= 3∫

X γ1 ∪ γ2 ∪ γ3 if n = 3

We conclude that

F(t, q)|q=0 =16

∫x(t0T0 + · · ·+ tmTm)

3.

The third partials Fαβγ := ∂3F∂tα∂tβ∂tγ

satisfy

Fαβγ

∣∣q=0 =

∫X

Tα ∪ Tβ ∪ Tγ.

And the Poincare pairing is just

F0βγ

∣∣q=0 = gβγ.

Note 1.19. Note that

Tα ∪ Tβ = ∑ε,ε′

Fαβε

∣∣q=0 gεε′Tε′ .


Definition 1.20. We define a product ∗ on H∗(X)⊗Q[[q1, · · · , qp, t0, · · · , tm]]by the formula

Tα ∗ Tβ = ∑ε,ε′

Fα,β,εgεε′Tε′

This is obviously commutative and when q = 0 then the product is0.Theorem 1.21. The product ∗ is an associative product.

In other words since

(Tα ∗ Tβ) ∗ Tγ = (Fαβεgεε′Tε′) ∗ Tγ

= Fαβεgεε′Fε′γδgδδ′Tδ′

(Tβ ∗ Tγ) ∗ Tα = Fβγεgεε′Fε′αδgδδ′Tδ′

which means that for all α, β, γ, and δ we have the WDVV equations

Fα,β,εgεε′Fε′γδ = Fβγεgεε′Fε′ ,α,δ.

These are partial differential equations for F. Before we prove WDVVlet’s see what F looks like in some examples. First we look at a fewsimple relations:

String equation: Note that 〈γ1, · · · , γn, T0〉0,β = 0 unless β = 0, n =

2. This is because

〈γ1, · · · , γn, T0〉β =∫[M0,n+1(X,β)]vir

ev∗1(γ1) ∪ · · · ∪ ev∗n(γn) ∪1︷︸︸︷

ev∗n+1(T0)

The forgetful map

M0,n+1(X, β)→M0,n(X, β)

exists as long asM0,n(X, β) is non-empty. So the integrand pulls backfromM0,n which has smaller dimension!

So what this implies about our potential function is that

∂F∂t0

= ∑β

〈exp(γ)T0〉βqβ =12〈γ2T0〉0,0

=12

∫X

γ2 =12 ∑

i,jtitjgij

F0αβ = gαβ so T0 = id for the product ∗.Divisor equation: For all i ∈ {1, · · · , p} and Ti ∈ H2(X),

〈γn, Tu〉0,β = (Ti.β︸︷︷︸βi

)〈γn〉0,β

unless β = 0 and n = 2.

∂F∂ti

= ∑β

〈exp(γ)Ti〉βqβ

= βi ∑〈exp(γ)〉βqβ +12〈γ2, Ti〉0,0


So

∂F∂ti

= qi∂F∂qi− 1

2

∫X

γ2Ti (1.3)

known as the Divisor equation. This implies that(∂

∂ti− qi

∂

∂qi

)Fαβγ = 0

expect for cubic terms in t when q = 0. The dependence of F on ti andqi is as a function of qieti . We know are going to attempt to finding Fin some cases.Example 1.22. First consider the case of X = P1. Here T0 = 1 andT1 = [pt]∨. In β = 0 the only non-zero invariant is 〈1, 1, [pt]∨〉 = 1. Inβ = d 6= 0 we have

〈T`1 〉d = d`〈.〉d =

{0 d 6= 11 d = 1

.

And vdimM0,0(P1, d[P1]) = −KX β + (dim X− 3)(1− g) = 2d− 2.

We conclude that

F =12

t20 t1 + q1 ∑

`

1`!〈T`

1 〉 t`1 =12

t20 t1 + q1et1 .

Jan 24

Example 1.23. Our second example is the case of X = P2. Recall thestring equation says 〈T0, γ1, · · · , γn〉0,β = 0 unless n = 2 and β = 0and that

〈Tα, Tβ, Tγ〉0,0 =∫

XT1α ∪ Tβ ∪ Tγ.

As a basis H∗(P2) take T0 = 1, T1 = H∨ where H ⊆ P2 is an embed-ded projective line and T2 = pt∨. The paring matrix is

(gij) =

0 0 10 1 01 0 0

.

F =t0t2

12

+t20t2

2+

∞

∑d=1

∑n1,n2≥0

〈ptn2 , Hn1〉0,dqd tn!1

n1!tn22

n2!

Here n2 = 3d− 1 and by divisor equation we can pull Hn1 out and get

F =t0t2

12

+t20t2

2+

∞

∑d=1

qd∞

∑n1≥0

dn1 〈pt3d−1〉0,d︸︷︷︸Nd

tn!1

n1!t3d−12

(3d− 1)!

=t0t2

12

+t20t2

2+

∞

∑d=1

qdedt1 Ndt3d−12

(3d− 1)!


Notice how appearance of qdedt1 is a common behavior is divisor equa-tion. Now we look at WDVV equations

Fαβεgεε′Fε′βδ = Fβγεgεε′Fε′αδ

And recalling what (gij) is here we get

F110︸︷︷︸1

F222 + F111F122 + F112 F022︸︷︷︸0

= F120︸︷︷︸0

F212 + F121F112 + F122 F012︸︷︷︸0

simplifying toF222 = F2

112 − F111F122.

Taking partials of the Gromov-Witten potential now tells us

F222 = ∑d(qet1)dNd

t3d−42

(3d− 4)!

F112 = ∑d

d2(qet1)dNdt3d−22

(3d− 2)!

F111 = ∑d

d3(qet1)dNdt3d−12

(3d− 1)!

F122 = ∑d

d(qet1)dNdt3d−32

(3d− 3)!.

Hence WDVV is

∑d>1

Ndt3d−4

(3d− 4)!= ∑

d1,d2>0Nd1 Nd2

(d2

1d22t3(d1+d2)−4

(3d1 − 2)!(3d2 − 2)!−

d31d2t3(d1+d2)−4

(3d1 − 1)!(3d2 − 3)!

)

This means that Nd is (3d− 4)!(t3d−4 coefficient of q) resulting Kont-sevich’s surprising result

(Kontsevich’s Formula)Nd = ∑d1+d2=dd1,d2>0

Nd1 Nd2

(d2

1d22

(3d− 43d1 − 2

)− d3

1d2

(3d− 43d1 − 1

))(1.4)

Let’s finish this discussion with a sketch of the proof of WDVVequation in the case whereM0,n(X, β) are smooth and of the expecteddimension at least in the example of X = Pn (where we do not need avirtual class).

Consider the following integral

(∗)n,α,β,γ,δ :=1n!

∫[M0,n+4(X,β)]

ρ∗(pt∨) ev∗1(Tα) ev∗2(Tβ) ev∗3(Tγ)

ev∗4(Tδ) ev∗5(γ) · · · ev∗n+4(γ)

where

ρ :M0,n+4(X, β)→M0,4∼= P1

[ f : (C, x1, · · · , xn+4 → X] 7→ (C, x1, · · · , x4)stab


We are assuming everything is smooth and ρ is transverse so we maywrite

ρ∗(pt∨) = (ρ−1(pt))∨.

We can choose the point pt to be any point inM0,4:

∈ M0,4.

So now we try to understand the integral ∗ in these terms:

(∗) = 1n!

∫ρ−1(pt)

ev∗1(Tα) ∪ · · · ∪ ev∗n+4(γ).

So

ρ−1

=

f : → X :

n1 + n2 = nβ1 + β2 = β

⊆

⊔n1+n2=n

β1+β2=β

M0,n1+3(X, β1)×M0,n2+3(X, β2)

where the disjoint union index n really ranges over all partitions {5, · · · , n+

4} = A1 ∪ A2 where |Ai| = ni. So

ρ−1

=(

L ev∗n1+3×∪ R ev∗n2+3

)−1(∆)

where ∆ ⊂ X× X is the diagonal, and dually

ρ∗

∨

=(

L ev∗n1+3×∪ R ev∗n2+3

)∗(∆∨)

but in any manifold ∆∨ = Tε ⊗ Tε (suppressing the summation nota-tion over ε hence

ρ−1

= L ev∗n1+3(Tε) ∪ R ev∗n2+3(Tε).


So the integral is

(∗) = 1n! ∑

n1+n2=nβ1+β2=β

(nn1

) ∫[M0,n1+3(X,β1)×M0,n2+3(X,β2)]

L ev∗n1+3(Tε) ∪ R ev∗n2+3(Tε)

L ev∗1(Tα) ∪ L ev∗2(Tβ)R ev∗1(Tγ) ∪ R ev∗2(T

δ)

L ev∗3(γ) · · · L ev∗n1+2(γ)R ev∗3(γ) · · · R ev∗n2+2(γ)

= ∑n1+n2=n

β1+β2=β

1n1!n2!

〈γn1 TαTβTε〉0,β1〈γn2 TγTδTε〉0,β2

We conclude that

∑n,β

(∗)qβ = ∑β1,β2

〈exp(γ)TαTβTε〉0,β1 qβ1〈exp(γ)TγTδTε′gεε′〉0,β2 qβ2

= Fα,β,εFγδε′gεε′

We therefore conclude that this is symmetric in α, β, γ and δ. Thisis very interesting, in fact the existence of a quantum product on thestructure of any manifold is interesting!

Jan 27

1.5 Gromov-Witten invariants of Calabi-Yau manifolds

We discussed the Gromov-Witten potential in previous lectures whichwas a big generating function with variables tαs and qis tracking in-sertions and track degree. We return now to Calabi-Yau threefolds inwhich case we recall that vdimMg(X, β) = 0. All information comesat t = 0.Definition 1.24. The genus g Gromov-Witten potential of a Calabi-Yauthreefold X is 1 1 The notation changed from q to v

through the history of these invari-ants as use of q became more promi-

nent in Donaldson-Thomas theory.

Fg = ∑β

NGWg,β vβ.

This series lives in Q[[v1, · · · , vp]] where vβ = vβ11 · · · v

βpp . Usually

this includes β = 0 when g ≥ 0 but sometimes this sum is written overβ 6= 0.Definition 1.25. The all-genus potential function is the formal Laurentseries

F = ∑g

Fgλ2g−2

in λ. The variable λ is known as string coupling constant.One more piece of definition is

Definition 1.26. The Gromov-Witten partition function is defined as

Z = exp(F) and Z0 = exp(F′) =exp(F)

exp(F|v=0)=

ZZ|v=0

.


Homework 1.27. Show that if we write

Z = ∑χ

∑β

N•χ,βλ−2χvβ.

Then N•χ,β can be interpreted as disconnected Gromov-Witten invari-ants:

N•χ,β =∫[M•χ(X,β)]vir

1

where

M•χ(X, β) =

f : C → X,possibly disconnected and/or nodalcurves such that f∗[C] = β and

χ(OC) = χ and |Aut( f )| < ∞

.

where χ(OC) is the number of components minus the sum of genera ofthe components. You may assume the following reasonable behaviorfor the virtual funcdamental class, that1. [M1 tM2]

vir = [M1]vir + [M2]

vir

2. deg[M1 ×M2]vir = deg[M1]

vir. deg[M2]vir

3. deg[M/G]vir = 1|G| deg[M]vir where G is a finite group.

Unlike genus-zero Gromov-Witten theory of P2 the relationship be-tween enumerative geometry (i.e. curve counting) and Gromov-Witteninvariants is more subtle.Example 1.28. X3

(5) ⊆ P4 generic quintic threefold. This has 2875 lineson it. They are all homologous and they generate H2(X, Z). They allhave normal bundle NP1/X

∼= O(−1)⊕O(−1). Note that since we arein a Calabi-Yau manifold, the degree of the normal bundle (here 2)is the same as the canonical bundle of P1 as is the case here. Everydegree 2 curve in X is a smooth plane conic. There are 609250 of them.There are no curves of genus strictly greater than zero in degrees 1 and2.

So what are the corresponding Gromov-Witten invariants. Let uswrite d for, d[`], i.e. d-times class of a line. And then the invariants weare looking for are denoted by Ng,d.

Every stable map of degree 1, genus zero is an isomorphism on toits image:

N0,1 = 2875

stable maps of degree 2, and genus zero are either isomorphismsonto one of the conics or a degree two multiple cover of a line. Thesecond factor is the contributions of multiple covers of lines.

N0,2 = 609250 + 1/82875

In factM0(X, 2) =

⊔609250

M0(P1, [P1])

⊔2875M0(P1,2[P1])


But notice that the second types of strata have dimension two which isbigger than the expected dimension. Hence it is a complicated prob-lem to understand how to account for its contribution.

In higher genera, for instance we have

N1,1 = 1/12(2875)

becauseM1(X, 1) =

⊔2875M1(P

1, [P1])︸︷︷︸P1×M1,1

.

1.6 Multiple cover formula–local P1

The fundamental problem here is what is the contribution of an iso-lated curve of genus g and degree β in X to Ng+h,dβ? Notice thatGromov-Witten invariants are not just well-defined for algebraic vari-eties but also symplectic manifolds. And in this general frameworkthese embedded curves are isolate.

For example if P1 ⊆ X with NP1/X∼= O(−1) ⊕ O(−1) then this

P1 is super-rigid in X, meaning there are no deformations (even in-finitesimal) of maps to P1 which deform off of P1. In other wordsMg(P1, d[P1]) lives as a component of the moduli space:

Mg(P1, d[P1]) ⊆Mg(X, d[P1])

and we can restrict [Mg(X, d[P1])]vir to this component and we get awell-defined contribution

NGWg,d ( local P1) =

∫[Mg(X,d[P1])]vir |Mg(P1,d[P1 ])

1

This is in other words, the universal multiple cover (or degeneratemap) contributions.

The conifold singularity is Spec C[x, y, z, w]/(xy − zw), which is athree dimensional variety with a singularity at the origin. This has aresolution (called conifold resolution)

X ∼= tor(O(−1)⊕O(−1)→ P1)

This is a famous model in Gromov-Witten theory and string theoryand is sometimes called the local-P1 .

We want to find the potential and partition function of this space:

NGWg,d (X) =

∫[Mg(X,d[P1])]vir

1 =∫[Mg(X,d[P1])]vir

eu(Ob)

the integrand is the Euler class of the obstruction sheaf Ob→Mg(P1, d[P1]).In particular,

Ob| f :C→P1 = H1(C, f ∗NP1/X) = H1(C, f ∗(O(−1)⊕O(−1)))


when C is smooth.This is computed via (virtual) localization. This technique uses C∗

action on Mg(P1, d[P1]) induced by the action of C∗ on P1. Thisreduces the integration to an integral overMg(P1, d[P1])C∗ . And thismoduli space can be explicitly described in terms ofMg1,n ×Mg2,n.

Mg(P1, d[P1])C∗ =

: g1 + g2 = g

We will learn how to do localization and will be able to calculate thisintegral. We will see that localization will reduce the problem to anintegral over

⊔g1+g2=gMg1,n ×Mg2,n. Eventually we will see that

Ng,d( local P1) = d2g−3 ∑g1+g2=g

bg1 bg2

by Bernoulli numbers that are defined for instance via

∞

∑g=0

bgt2g =

(sin(t/2)

t/2

)−1

The potential function will therefore be

F = ∑g,d:d 6=0

Ng,dλvd = ∑g1,g2≥0,d>0

dbg1 bg2 λ2g1+2g2−2vd

= ∑d>0

vdd−3λ−2

(∑g1

bg1 d2g1 λ2g1

)(∑g2

bg2 d2g2 λ2g2

)

= ∑d>0

vdd−3λ−2(

sin(dλ/2)dλ/2

)−2

= ∑d>0

vd

d(2 sin(dλ/2))−2 .

In genus zero, F0 is λ−2-term of the above which is

(Aspinwall-Morrison Formula)F0 = ∑d>0

vd

d3 (1.5)

This 1d3 is the multiple cover contribution in genus zero.

Jan 29

1.7 Multiple cover formula

Recall that we have been discussing contributions from multiple cov-ers and degenerate contributions. The general idea, is to simplify


Gromov-Witten theory of Calabi-Yau threefolds by systematically re-moving multiple cover contributions.

Last time we began to execute this idea if P1 ⊆ X with NP1/X∼=

O(−1)⊕O(−1). We observed that the contribution of this P1 to NGWg,d[P1]

is well-defined because

Mg(P1, d[P1]) ⊂Mg(X, d[P1])

as an isolated connectedness component. In other words not just thatthese maps from P1 do not deform in families but they do not deforminfinitesimally either. In general we haveDefinition 1.29. A curve C ⊂ X is super-rigid, if

Mg(C, d[C]) ⊆Mg(X, d[C])

as a scheme-theoretic connected component. Then we define

NGWg,d ( local C) =

∫[Mg(X,d[C])]vir |Mg(C,d[C])

1.

Note that super-rigidity is easily seen to be satisfied in the symplec-tic world. But is a more subtle condition here. However we still expectit to be a reasonable criterion.Example 1.30. Last time we derived that NGW

g,d ( local P1) has potentialfunction

F′ = ∑d>0,g

Ng,dλ2g−2vd = ∑d>0

vd

d

(2 sin

(dλ

2

))−2.

Hence F0 = ∑d1d3 vd which is the simplest of multiple-cover formulae.

Note that although this looks like a nice formula, it is actually prettycomplicated; every P1 is contributing in every degree, and in everygenus. So these types of formulae beg for some unraveling.Example 1.31. Suppose E ⊂ X is a smooth elliptic curve with NE/X

∼=L⊕ L−1 where L is a degree zero line bundle. And since we are talkingabout elliptic curves we have L ∈ Pic0(E). A line bundle is non-trivialis it does not have any non-zero sections. But if we want it to not haveany multi-sections then it is non-torsion, i.e. L⊗n 6∼= OX for any n. ThenE ⊆ X is super-rigid so we get

NGWg,d ( local E) =

∫[Mg(X,d[E])]vir |Mg(X,d[E])

1.

And one can use degeneration techniques to prove the followingTheorem 1.32. For genera other than 1, g 6= 1 we have for an elliptic curve

NGWg,d ( local E) = 0.


Homework 1.33. Use the covering space theory to show that thereare σ(d) = ∑k|d k (connected) covering spaces of degree d over E andeach is a normal covering space with cardinality of the group of decktransformations being d

So what does this tell a Gromov-Witten theorist? To compute NGW1,d ( local E)

we need to studyM1(E, d[E]). By Riemann-Hurwitz formula f : E→E is unramified with smooth domain. Thus

M1(E, d[E]) = σ(d) isolated points

and each of these points is an orbifold point of the form [∗/G] with|G| = d. And this implies that the fundamental class is just the sum ofthese points devided by order of the group. So eventually we concludethat these invariants are

NGWg,d ( local E) =

{0 g 6= 1σ(d)

d g = 1.

So we have

F′ = ∑d>0

N1,dvd = ∑d>0

1d

σ(d)vd

=∞

∑d=1

1d ∑

k|dkvd = ∑

k,m≥1

kkm

vkm

=∞

∑k=1

∞

∑m=1

1m(vk)m =

∞

∑k=1− log(1− vk)

So we have

F′ = log(∞

∏k=1

1(1− vk)

).

And

Z′ = exp(F′) =∞

∏k=1

1(1− vk)

=∞

∑n=0

p(n)vn.

where p(n) is the number of partitions of n, i.e. ways of writing n =

λ1 + λ2 + · · ·+ λ`, with λ1 ≥ λ2 ≥ · · · ≥ λ` > 0.

Figure 1.11: Lattice picture

Figure 1.12: The map contribut-ing to disconnected invariants

Recall that by homework 1.27, Z′ is the generating function for (pos-sibly) disconnected Gromov-Witten invariants. So p(n) is really thedegree n, χ = 0 disconnected local Gromov-Witten invariant of E, i.e.it is the number of possible disconnected unramified covers of a toruscounted with automorphisms.Example 1.34. For example with d = 2, we have σ(2) = 3 we cansee this from figure 1.11. And p(2) the cardinality of set of partitions,{1 + 1, 2} so p(2) = 2.

Imagine . . . we might suppose there are some underlying integersng,β(X) (some sort of enumerative count) such that {NGW

g,β (X)} can berecovered from ng,β(X) by applying multiple-cover formulae.


Conjecture 1.35 (Gupakumar-Vafa). There exists ng,β(X) ∈ Z such thatwe may write

F′(X) = ∑g≥0,β 6=0

NGWg,β (X)λ2g−2vβ

= ∑g≥0,β 6=0

ng,β(X)∞

∑k=1

1k

(2 sin

(kλ

2

))2g−2vkβ. (1.6)

And moreover, ng,β(X) = 0 for g > c(β).For example our formula implies that

ng,d( local P1) =

{1 d = 1, g = 00 otherwise.

and

ng,d( local E) =

{1 g = 1, all d0 otherwise.

Gupakumar and Vafa gave a physics definition of ng,β(X). But untilrecently there has not been any satisfactory geometric definition. Nev-ertheless we can take the formula 1.6 as the definition of {ng,β(X)} interms of {NGW

g,β (X)}. The content of the conjecture is then the integral-ity and finiteness conjecture

ng,β ∈ Z and ng,β = 0 for g > c(β)

and in fact this version of the conjecture has been proven after 15 yearsin 2013 using symplectic geometry: Parker and Ionel showed that us-ing almost-complex structures this picture is not too far off. The localtheory was proved by Bryan and Pandharipande. And we will talkabout this later.

In some sense Gupakumar-Vafa is the most fundamental invariantsamong the ones we are investigation. But the issue with them is thatunlike the other invariants it is hard to invent a rigorous mathematicalframework for it.

Chapter 2Donaldson-Thomas Theory

2.1 Moduli of ideal sheaves

Let In(X, β) be the Hilbert scheme of subschemes C ⊆ X with [C] =β ∈ H2(X, Z) and χ(OC) = n. Here we use this non-conventionalnotation for Hilbert schemes because we really want to think aboutthis as a space of ideal sheaves. The subscheme can be non-reduced,reducible and even can have zero dimensional components. In the casethat C ⊆ X is smooth and connected then of course χ(OX) = 1− g.

The same sort of anomalies that occurred in Gromov-Witten theorycan also happen here as well. Recall that a smooth conic could bedegenerating to a double line, and the map used to become a doublecover. And now here we can also have that the image of such map is anon-reduced curve in the class β with length 2 scheme structure (figure2.1). Here this scheme locally is Spec C[x, y]/(x2). The analogue ofcollapsing components here are embedded points on a scheme. Forexample in figure 2.2 the scheme locally is Spec C[x, y]/(xy, x2).

Figure 2.1: Analogueof degenerate maps

Figure 2.2: Analogue of collapsingcomponents are embedded points

Also subschemes can be disconnected. So unlike Gromov-Wittentheory, Donaldson-Thomas theory in inherently disconnected. How-ever as we saw in homework 1.27 we can have a disconnected versionof Gromov-Witten theory that turn out to have subtle connections withDonaldson-Thomas theory.

What can happen in a flat family of subschemes? But in cases ofprojective families there is an easy equivalent condition for the familyCt ⊆ X of curves being flat; that of, [Ct] = β and χ(Ct) = n beingconstant in the family.Example 2.1. Let us look at a family of curves in P3. P3 is not a Calabi-Yau threefold but this example serves to illustrate the phenomena. Solet Ct ⊆ P3 be a family of twisted cubics degenarating to a plane cubic.The curve P1 ↪→ P3 via (x : y) 7→ (x3 : x2y : xy2 : y3) has genus zero.


But it degenerates to a curve in the plane, with a node. But a reducedplane cubic C with a node has χ(OC) = 1− g = 0 by Riemann-Rochagain. So there must have been an embedded point at the node (figure2.3).

Figure 2.3: Degeneration of the cubic(x, y) 7→ (x3, x2y, xy2, y3) in P3 to re-

duced plane cubic curve C with a node

The local model is

Ct 6=0 = {z = x = 0} ∪ {y = z− t = 0} ⊆ C3

with has ideal (x, z)(y, z− t) = (xy, yz, xz− xt, z2 − zt). The flat limitas t → 0 is (xy, yz, xz, z2). However (xy, yz, xz, z2) ( (z, xy) where thelatter is the reduced structure on the locus of the degeneracy (figure2.4).

Figure 2.4: The localmodel of the degeneration

And this type of behavior comes as a surprise because furthermore,this family can degenerate into a nodal curve union an isolated point,and further into a smooth genus 1 curve with a point.

Figure 2.5: Possible deformations infamilies of curves. The limit can be asmooth curve with an isolated point.

For deformation theoretic reasons it is crucial that we regard In(X, β)

as not parametrizing subschemes but we want to regard it as parametriz-ing sheaves. In fact, let I be a stable sheaf on a Calabi-Yau threefold(and for simplicity we consider H1(X, OX) = 0 in definition of a Calabi-Yau) with Chern character

ch(I) = (1, 0,−β,−n) ∈ H0 ⊕ H2 ⊕ H4 ⊕ H6

then the mapping I → I∨∨ is an embedding I ↪→ I∨∨ ∼= OX by stability.Hence I is an ideal sheaf of some subscheme C ⊆ X with β = [C] andχ(OC) = n. This tells us why we may really forget about subschemesand think of objects of our concern as sheaves. (We could have forgotabout this embedding of sheaves, and consider sheaves I with fixeddeterminant).

For curve counting invariants we use In(X, β) moduli space of idealsheave [MNOP]. But Donaldson-Thomas theory is actually defined forany moduli space of stable sheaves. In fact, the original interest ofDonaldson and Thomas were bundles.

2.2 Digression to deformation theory of bundles

There are different approaches to deformation theory. One is a purelyalgebraic approach which is too complicated for our purposed. We

donaldson-thomas theory 35

will consider the complex geometric approach which is very intuitiveand works pretty easily for case of bundles.

Suppose E→ X is a bundle given by transition functions

gαβ : Uα ∪Uβ → Gln(C).

Recall that this data has to satisfy the co-cycle condition

gαβgβγgγα = 1 on Uα ∪Uβ ∪Uγ,

in other words, {gαβ} ∈ H(X, GLn(OX)).A deformation of E if given by {gαβ(t)} such that

gαβ(t)gβγ(t)gγα(t) = 1 on Uα ∪Uβ ∪Uγ, and gαβ(0) = gαβ.

An infinitesimal deformation of E is {gαβ(t)} such that

gαβ(t)gβγ(t)gγα(t) = 1 mod t2 on Uα ∪Uβ ∪Uγ, and gαβ(0) = gαβ.

Claim: Infinitesimal deformations of E are given by h ∈ H1(X, End(E)).let h = {hαβ} ∈ H1(X, End(E)). So

hαβ + hβγ + hγα = 0 on Uα ∪Uβ ∪Uγ.

But what does this condition mean? Unlike gαβ’s, the hαβ’s are notvalued in fixed matrices, but instead it is valued in End(E|Uα∩Uβ

). Toview hαβ is a matrix valued function, we need to choose a trivialization,by convention, we will write hαβ in the Uβ basis. Then hαβ in the Uα

basis is given by gαβhαβg−1αβ.

So cocycle condition written in the Uα basis reads

gαβhαβg−1αβ + gαγhβγg−1

αγ + hγα = 0.

Let gαβ(t) = gαβ(1 + thαβ + · · · ) We can calculate gαβ(t)gβγ(t)gγα(t)up to order one, and we get

gαβ(t)gβγ(t)gγα(t) = 1 + t(gαβhαβg−1αβ + gαγhβγg−1

αγ + hγα + o(t2).

We can push this idea further to determine which infinitesimal de-formations can be lifted to deformations of all orders. The obstructionslive in H2(X, End(E)) = Ob(E). The moduli space of bundles near ofE is given by the zero locus of a (non-linear!) map

H1(X, End(E))→ H2(X, End(E)).

Let M be the moduli space of (stable) bundles. Then T[E]M =

Def(E) = H1(X, End E). Locally M is the zero locus of

Def(E)︸︷︷︸H1(X,End(E))

→ Ob(E)︸︷︷︸H2(X,End(E))

.


2.3 Moduli spaces as critical loci of functionals

All this picture works as well in the algebraic context of coherentsheaves, with the deformation and obstruction spaces being replacesnow by

Def(E) ∼= Ext1(E, E)

Ob(E) ∼= Ext2(E, E)

Using Serre duality

Exti(F, G) ∼= Extdim X−i(G, F⊗ KX)∨.

in case that X is moreover a Calabi-Yau threefold we have Ext1(E, E) ∼=Ext2(E, E)∨, i.e. the spaces of deformations and obstructions are dual.

Feb 3

Last time we found that for E → X a holomorphic bundle on acomplex manifold, infinitesimal deformations of E are classified byH1(X, End(E)):

T[E]M = H1(X, End(E)),

where M is the moduli space of bundles.Obstructions to lifting infinitesimal deformations to infinite order

lie in H2(X, End(E)). Locally the moduli space is the zero locus of afunction

κ : H1(X, End(E))→ H2(X, End(E))

known as the Kuranishi map.Specific of Calabi-Yau threefolds is that by Serre duality,

H1(X, E⊗ E∗) ∼= H2(X, (E⊗ E∗)∗ ⊗ KX)∨

so the space of deformations is

Def = H2(X, End(E))∨ = (Ob)∨.

So we may view κ as a sectino of T∗H1(X, End(E))→ H1(X, End(E)).In other words, κ is a 1-form which is necessarily exact, κ = d f . Hencef is characterizing the moduli space M locally near [E] as the criticallocus of f ,

M ∼= crit( f ).

This map f : H1(X, End(E)) → C is know as the Chern-Simon (super-)potential.

A priori f is the germ of a holomorphic function, or in the algebraiccontext, a formal function. But recently is has been proven in casesthat it is realized as a polynomial.


More generally, if E is now a coherent sheaf and M is the modulispace of sheaves, then T[E]M = Ext1(E, E) and Ob(E) = Ext2(E, E).And the Kuranishi map now has an algebraic version

κ : Ext1(E, E)→ Ext2(E, E) ∼= Ext1(E, E)∨.

So locallyM∼= crit( f : Ext1(E, E)→ C).

2.4 Donaldons-Thomas invariants

IfM is a compact moduli space of sheaves on a Calabi-Yau threefolds,as in example of In(X, β), (this usually involves a choice of a stabilitycondition then [M]vir ∈ H0(M, Z) as hence we obtain a numericalinvariant

NDTM (X) =

∫[M]vir

1 ∈ Z.

In particular our curve counting invariants are

NDTn,β (X) =

∫[In(X,β)]vir

1

which should be thought of a virtual count of degree β curves C in Xwith χ(OC) = n.

Suppose M is a compact moduli space of sheaves on a Calabi-Yauthreefold, and suppose M is smooth of dimension d. Then the ob-struction sheaf Ob → M is a bundle of rank d, then Ob ∼= T∗M. Sothe virtual class is the top Chern class of the cotangent bundle:∫

[M]vir1 =

∫[M]

cd(T∗M) = (−1)d∫[M]

dd(TM) = (−1)de(M),

where e(M) is the top Euler character of M.Now the amazing fact is that this behavior as an Euler characteristic,

is true even in vaseM is not a smooth space:Theorem 2.2 (Behrend).∫

[M]vir1 = ∑

k∈Z

ke(ν−1M (k)) =: evir(M) (2.1)

where νM : M→ Z is the behrend function defined by

νM([E]) = (−1)dim Ext1(E,E)(1− e(MFf ))

where MFf is the Milnor fiber of the Chern-Simons superpotential, f .If [E] ∈ M is a smooth point ofM then

νM([E]) = (−1)dim Ext1(E,E).

Figure 2.6: Milnor fiber


Milnor fiber of f : Ext1(E, E)→ C is by definition

MFf = { f−1(δ) ∩ Bε : 0 < δ� ε� 1}.

As quick examples if M is smooth and f = 0 then MFf = ∅. If M =

Spec{C[x]/xn}, if the fat point of length n, in which case f : C→ C isx 7→ xn+1, then M = {d f = 0} = {xn = 0}.1 1 For simple spaces, structure of M

determines the potential f . But ingeneral one needs to investigate theKuranishi map to understand f . In

fact f is not uniquely determinedby M in general however one ad-vantage of Behrend’s function is

that it is uniquely determined by M.

The Milnor fiber in this example is

MFf = {xn+1 = δ} ∩ Bε ∼= {xn+1 ∼= 1}.

Let us compact the different paradigms the two sides of formula2.1. The left hand side is an integral over a moduli space, whereas theright hand side is the euler characteristic of some sort. In fact1. The right hand side is define even if M is non-compact (e.g. If X is a

non-compact Calabi-Yau threefold). So we may use this right handside, more generally as our definitions for invariants in the moregeneral context.

2. Euler characteristic can be computer motivicaly:

e(X) = e(X r Z) + e(Z) if Z ⊆ X is a closed set, and also

e(X1 × X2) = e(X1)e(X2).

3. By localization ifM admits a C∗-action then

e(M) = e(MC∗),

and this follows from the previous motivic property of these invari-ants.

4. Categorification: the ordinary euler characteristics are alternatingsums of Betti numbers: e(M) = ∑(−1)n dim Hn(M). This suggestsexistence of some cohomology theory H∗(M) such that evir(M) =

∑(−1)n dim Hn(M). Existance of H∗(M), implies a categorificationof Donaldson-Thomas invariants associated to M; that is a transitionfrom the set of numbers (where the numerical invariants live) to thecategory of vector spaces (where the cohomological invariants live).However, there is essentially one important feature that can be seen

on the left hand side the formula 2.1 and not the right hand side, andthat is deformation invariance!

Feb 5

Last time we saw that that we can use weighted topological Eulercharacteristics instead of virtual classes in Donaldson-Thomas theory.We talked about In(X, β), the Hilbert scheme of subschemes C ⊆ Xwhere [C] = β, n = χ(OX) and viewed is as moduli space of idealsheaves:

NDTn,β (X) =

∫[In(X,β)]vir

1 = evir(In(X, β)).


Example 2.3. We consider the local P2: X = tot(O(−3) → P2). Let` = [P1] be the class of a line. Then the lines on the base (figure 2.7)are the only curves that contribute to:

Figure 2.7: Lines in P2

I1(X, `) = { lines in P2 } ∼= P2.

So NDT1,` (X) = e(P2) = 3. Similarly, in degree two the conics (figure

2.8)

Figure 2.8: Conics in P2

I1(X, 2`) = { conics in P2 } = P5.

So NDT1,2`(X) = (−1)e(P5) = −6. In higher euler characteristics things

get more complicated:

I2(X, `) =

This space is more complicated. Firstly, unlike the previous items thisspace is non-compact. Although it is easy to see that it is 5 dimensional(the isolated point lives in the three dimensional total space and theline lives in the zero section). However, this space turns out to besmooth and hence

NDT2,` (X) = (−1)e(In(X, `)).

One technique for computing on this space is using a torus action. LetT = (C×)3 act on In(X, `) in this manner: we lift an action of a torusT on P2 to an action on O(−3)→ P2, i.e. on X, such that the action isnontrivial on the fibers of the fixed loci of P2. This induces an actionof T on In(X, d`).

Now for any space Y, with a torus action, e(Y) = e(YT). Therefore

In(X, d`)T = {T invariant subschemes C ⊆ X}.

So

I2(X, `)T =

6 types like

∪ 3 points like

.

And hence

NDT2,` (X) = (−1)(6)#

{Z ⊆ C3,

T-invariant, supported on z-axiswith embedded point at the origin

}− 3

= −3− 6.#

{I ⊆ C[x, y, z],

I is generated by monomials√I = (x, y), dim

√I/I = 1

}.


To do this count we visualize monomials as boxed in the positive oc-tant of the 3-space. Appearance of a monomial xiyjzk in generators, ac-counts to removing a shifted octant form the a configuration of boxes.Our favorite configurations are the ones that asymptotically only haveboxes along the z-axis. Fat points at the origin correspond to a cum-mulate of finitely many boxes in neighbourhood of the origin. Herewe only have two possibilities as shown in figure 2.9.

Figure 2.9: Configurations that asymp-totically have one box along the z-axis

respectively corresponding to (xz, x2, y) and (x, y2, zy). So we con-clude that NDT

2,` (X) = −3− 6.2 = −15. In the presence of a T-actionwe can compute NDT by box counting. We will do this systematicallywhen we study topological vertex.Remark 2.4. In Gromov-Witten theory, we also have a localization tech-nique. There we are not dealing with euler characteristics and thesituation is more complicated, namely that of Atiyah-Bott localization.For example,

M1(X, `)T = 6 copies ofM1,1

as hinted by in the figure 2.10). SoMg(X, d`)T can be written in termsofMg,d.

Figure 2.10: Atiyah-Bott localization

A few more examples:Example 2.5. Let X be the total space of O(−1) ⊕ O(−1) → P1 thistime.

Figure 2.11: The local P1

This space has a T-action with 2 fixed points on a fixed curve. Obvi-ously NDT

1,` (X) = 1. We have NDT2,` (X) = (−1).#I2(X, `)T = −4 counted

by configurations of the type shown in figure 2.12.

Figure 2.12: Configura-tions contributing to I2(X, `)

And NDT3,` (X) = #I3(X, `)T = 14 is counted by following types of

configurations (figure 2.13).

Figure 2.13: Configura-tions contributing to I3(X, `)


Feb 7

2.5 GW/DT correspondence

Recall that NDTn,β (X) =

∫[In(X,β)]vir 1 = evir(In(X, β)). The partition func-

tion for Donaldson-Thomas invariants is defined as

ZDTβ (X) = ∑

nNDT

n,β (X)qn ∈ Z[q−1][[q]].

Recall that

FGWg (X) = ∑

β

NGWg,β (X)vβ

(FGWg (X))′ = ∑

β 6=0NGW

g,β (X)vβ

FGW(X) = ∑g

FGWg (X)λ2g−2

(FGW(X))′ = ∑g(FGW

g (X))′λ2g−2

ZGW(X) = exp(FGW(X))

(ZGW(X))′ = exp(F′) = exp(F− F|v=0)

From homework we have seen an interpretation of the coefficients ofZGW(X) and (ZGW(X))′. In fact

ZGW(X)′ = ∑β,χ

N•,GWβ,χ (X)′vβλ−2χ.

hence N•β,X has a geometric interpretation too: it is the virtual countof possibly disconnected stable maps of degree β and χ(C) = χ suchthat no connected component collapses.

This suggests that we should be trying to relate ZGW(X)′ and ZDT(X)′.Here we define ZDT(X)′ as

ZDT(X)′ =ZDT(X)

ZDT0 (X)

.

This fraction has no obvious geometric interpretation. Loosely speak-ing we do not allow points away from curves.Conjecture 2.6 (MNOP).

ZDTβ (X)′ = ZGW

β (X)′

after the change of variables q = −eiλ.This is a priori a very bizarre change of variables! In fact for the

conjecture to even make sense we need


Conjecture 2.7. ZDTβ (X)′, a priori in Z[q−1][[q]] is a rational function in q

invariant under q↔ q−1, i.e. it is linear combination of q(1−q)2 , qk

(1−qk)2 , · · · .Because this change of variables is happening through analytic con-

tinuation, in order to prove this conjecture, we cannot compare invari-ants number by number but we need to get a single Gromov-Witteninvariant NGW

β,g we must know NDTβ,n for all n. To get a single Gromov-

Witten invariant NGWβ,g we must know NDT

β,n for all n.

Remark 2.8. ZGWβ (X)′ is an expansion about λ = 0 and ZDT

β (X)′ is anexpansion about q = 0, i.e. as λ → i∞. In physics jargon, this iscalled non-perturbative duality. ZGW

β (X) is the perturbative expansion

at weal string coupling and ZDTβ (X)′ is the perturbative expansion at

strong string couplings. These partition functions are energies of thevacuum in different theories. The Gromov-Witten partition functionis related to Type A topological string theory and the Donaldson-Thomas one is related to Type B topological string theory (with D0-D2-D6 branes). The relation between the two is known as S-duality.Example 2.9. Let us observe this rationality conjecture in case of theresolved conifold, a.k.a. the local P1:

X = tot(O(−1)⊕O(−1)→ P1).

Recall that

FGW(X)′ = ∑d>0

vd

d

(2 sin

(dλ

2

))−2.

So setting Q = eiλ we get that

FGW(X)′ = ∑d>0−vd

d((e−idλ/2)(eidλ − 1))−2

= ∑d>0−vd

dQd

(1−Qd)2

= ∑d>0−vd

d ∑m>0

mQmd

=∞

∑m=1

m∞

∑d=1− (vQm)d

d

=∞

∑m=1

m log(1− vQm)

So

FGW(X)′ = log

(∞

∏m=1

(1− vQm)m

).

And recall that

ZGW(X)′ =∞

∏m=1

(1− v(−q)m)m = ZDT(X)′

under our change of variables.


Part of the goal today is to examine this conjecture in a few cases.Example 2.10. The β = [P1]-term:

Z′[P1](X) = coefficient of v1 in∞

∏m=1

(1− v(−q)m)m

But we have

∞

∏m=1

(1− v(−q)m)m =∞

∏m=1

(1−mv(−q)m + O(v2))

= 1−∞

∑m=1

mv(−q)m + O(v2)

So

Z′[P1](X) =∞

∑m=1−m(−q)m = q− 2q2 + 3q3 + · · · .

So the degree-one prediction here is that

q− 2q2 + 3q3 − · · · =∑∞

n=1 NDTn,[P1]

(X)qn

∑∞n=0 NDT

n,0 (X)qn.

But we already know that

NDTn,0 (X) = evir(Hilbn(X))

NDT0,0 (X) = 1 the “ideal” sheaf in OX

NDT1,0 (X) = evir(X) = −e(X) = −2.

NDT1,[P1](X) = evir(I1(X, [P1])) = 1

NDT2,[P1](X) = evir(I2(X, [P1])) = (−1) e(I2(X, [P1])T)︸︷︷︸

4 fixed points of torus action

So the conjecture predicts that

(q− 2q2 + 3q3 − · · · )(1− 2q + · · · ) = (q− 4q2 + · · · )

It is already miraculous that the first factor on left hand side has in-teger coefficients. The fact that the two series start both at the sameterm is already also miraculous from point of view of Gromov-Wittentheory.

To use the torus action to compute evir(In(X, β)) we need to knowthe value of Behrend function at the fixed points of the action.

Feb 12

Suppose X is a toric Calabi-Yau threefold. Such a variety is a T =

(C×)3 equivariant partial compactification of (C×)3. So T acts on Xwith isolated fixed points. And there are T-equivariant coordinatesabout each fixed point.


The induced action of T on In(X, β) has isolated fixed points cor-responding to subschemes Z ⊂ X which are T-invariant. These aredefined by monomial ideals on each coordinate patch. And are alsoin one-to-one correspondence with “piles of boxes” in the positive oc-tant. There is a 2-dimensional subtorus T′ ⊂ T which acts trivially onKX = X×C. It is a little harder to see that

In(X, β)T = In(X, β)T′ .

For fixed points of such group actions, the group acts on the tangentspace. So if [I] ∈ In(X, β)T′ , then T′ acts on Def(I) = Ext1(I, I) and onOb(I) = Ext2(I, I) and the Kuranishi map κ is T′-equivariant:

κ : Ext1(I, I)→ Ext2(I, I).

Remember that κ = d f for f the Chen-Simons potential f : Ext1(I, I)→C. This implies that f is T′-invariant. The claim is that this implies thate(MFf ) = 0.

Recall that ν([I]) = (−1)dim Ext1(I,I)(1− e(MFf )). Here

MFf = { f−1(δ) ∩ Bε : 0 < δ� ε� 1}.

We may fix a Hermitian metric on Ext1(I, I) (we need to do that any-ways to have the balls defined by that metric but) in such a way thatS1 × S1 ⊂ T1 ∼= C× × C× acts unitarily. Then S1 × S1 acts freely onMFf . This implies e(MFf ) = 0. Consequently ν([I]) = (−1)dim Ext1(I,I).Proposition 2.11 (MNOP). Let I ∈ In(X, β)T , then dim Ext1(I, I) mod 2,depends on β and linearly on n mod 2.

This implies that

ZDTβ (X) = ∑

nevir(In(X, β))qn = ±∑

n(−1)n#In(X, β)Tqn.

Example 2.12. Let x1, · · · , xF be the fixed points of X. And recall thatlocally Hilbn

xi(X) is the same moduli space as Hilbn

0 (C3). So,

ZDT0 (X) =

∞

∑n=0

(−1)n# Hilbn(X)Tqn

= ∑n

∑n1+···+nF=n

F

∏i=1

(Hilbni

xi (X)T(−q)ni)

=

(∑n

# Hilbn0 (C

3)(−q)n

)F

= M(−q)F.

Here M(q) = ∑∞n=0(# 3d-partitions of n)qn = ∏∞

m=1(1 − qm)−m. No-tice also that F is the topological euler characteristic of X, hence weconclude that Z0(X) = M(−q)e(X).


Theorem 2.13. Z0(X) = M(−q)e(X) even if the Calabi-Yau threefold X isnot toric.Example 2.14. X = tot(O(−1)⊕ O(−1) → P1). The prediction fromGromov-Witten theory was that

ZDT(X)

ZDT0 (X)

=∞

∏m=1

(1− v(−q)m)m.

The v-term is

ZDT1 (X)

ZDT0 (X)

=q

(1 + q)2 = q− 2q2 + 3q3 − 4q4 + · · · .

The MacMahon function function is M(q) = q+ 1+ 3q2 + 6q3 + 14q4 +

· · · And one can check that the identity

q− 4q2 + 14q3 − 42q4 + · · ·(1− q + 3q2 − 6q3 + · · · )2 = q− 2q2 + 3q3 + 4q4 + · · ·

holds in the first few terms.Feb 14

2.6 Topological vertex

For same space we now want to compute Donaldson-Thomas invari-ants by counting In(X, d[P1])T , i.e. counting subschemes by monomialideals in each coordinate patch, and i.e. box configurations of the fol-lowing form. [Z] ∈ In(X, d[P1])T , such that Z ∩ C2 ⊂ C2 of length dis invariant under (C×)2-action on C2 where C2 is considered is thegeneric fiber of X → P1. In other words it is given by a 2D partition λ

of d.Recall that λ ` d if λ = (λ1, · · · , λ`) is a sequence of positive inte-

gers satisfies λ1 ≥ λ2 ≥ · · · ≥ λ` and |λ| = ∑`i=1 λi = d. We picture

this as d boxes in the positive quadrant as in figure 2.14 is such away that C[x, y]/Iλ has a basis as a C-vector space labelled by every� ∈ λ. So for instance in the above picture we have λ = (5, 4, 1, 1) andIλ = (y5, xy4, x2y, x4).

Figure 2.14: Interpretation of a2D-partition as a quotient ring.

Let Zλ ⊂ X be the T-fixed subscheme with cross-section given by λ

and no embedded points. In general In(X, d[P1])T parametrizes sub-schemes that look like Zλ for some λ ` d with embedded points at0 and ∞. In each coordinate chart our subscheme is determined by a3D partition in the positive octant, asymptotic to λ in the z-direction(figure 2.15).

There exists a combinatorial tool that counts these configurations ofboxes and it is called the topological vertex:


Definition 2.15. V∅∅λ(q) = ∑π q|π| where the summation ranges oversall 3D partitions that are asymptotic to λ and |π| is the number ofadded boxes (i.e. the orange boxes in pictures like above).

Figure 2.15: The asymptote of λ hascontains three cubes in z-direction.

Theorem 2.16 (Okounkov, Reshetikhin, Vafa).

V∅∅λ(q) = M(q)q−(λ2)Sλt(1, q, q2, q3, · · · ).

Here M(q) is the usual MacMahon function

M(q) =∞

∏m=1

(1− qm)−m = V∅∅∅(q)

and the notation (λ2) is really the summation ∑i (

λi2 ). λt is the transpose

of λ defined in the obvious way.And Sλt(x1, x2, · · · ) is the Schur function labelled by λ. It is a sym-

metric function homogeneous of degree |λ|. And Sλ for a basis forsymmetric function. In particular if λ ` 1 then λ = � and

S�(x1, x2, · · · ) = x1 + x2 + · · ·

henceS�(1, q, q2, · · · ) = 1 + q + q2 + · · · = 1

1− q.

In fact it turns out that Schur functions are all rational. One usefulidentity that holds for Schur functions is

∑λ

Sλ(x1, x2, · · · )Sλt(y1, y2, · · · ) = ∏i,j(1 + xiyj).

Since V∅∅λ = V∅∅λt then

q−(λ2)Sλt(1, q, q2, · · · ) = q−(

λt2 )Sλ(1, q, q2, · · · ).

In fact this is equal to ∏�∈λ1

1−qh(�) where h(�) is the number of boxes

in a hook.Theorem 2.17 (MNOP). For any [IZ] ∈ In(X, d[P1]) we have

dim Ext1(IZ, IZ) = d + n mod 2

and consequently, ν([IZ]) = (−1)d+n.Calculation in same reference shows that

χ(OZλ) = |λ|+

(λ

2

)+

(λt

2

).

So we can now compute the DT partition function

ZDT(X) =∞

∑d=0

∑n

NDTn,d[P1](X)vdqn

=∞

∑d=0

(−1)dvd ∑λ`d

V∅∅λ(−q)V∅∅λ(−q)(−q)|λ|+(λ2)+(λt

2 )


By a change of variables we have

ZDT(X)(−q,−v) = ∑λ

v|λ|V∅∅λ(q)2(q)|λ|+(λ2)+(λt

2 )q−2(λ2)

= M(q)2︸︷︷︸ZDT

0 (X)(−q)

∑λ

v|λ|Sλ(q)Sλ(q)q|λ|q(λ2)q(

λt2 )

And remember we were really interested in the reduced Donaldson-Thomas theory:

ZDT(X)(−q,−v)′ =ZDT(X)(−q,−v)

ZDT0 (X)(−q)

= ∑λ

(qv)|λ|Sλ(q)Sλt(q)

= ∑λ

Sλ(qv, q2v, q3v, · · · )Sλt(1, q, q2, · · · )

= ∏i,j≥1

(1 + vqi+j−1)

=∞

∏m=1

(1 + vqm)m.

And we conclude that

ZDT(X)′ =∞

∏m=1

(1− v(−q)m)m

as predicted.

Chapter 3Pandharipande-ThomasTheory

Feb 26

Let’s recall some of the basic problematic phenomena in Donaldson-Thomas theory. We saw that smooth genus g curves can deform infamilies to nodal curves with embedded points and further to smoothgenus g + 1 curves together with a point:

The Hilbert scheme aka the moduli space of ideal sheaves includesfloating points! This is undesirable for curve counting. In fact becauseof this, the MNOP conjecture needed to remove point contributionsformally via

ZDT(X)′ =ZDt(X)

ZDT(X)|v=0= M(−q)−e(X)ZDT(X).

We want a theory that revives the right hand side M(−q)−e(X)ZDT(X)

somehow “on its own”.So the question of better compactification is: are there limits that

do not create embedded points? For instance, let Ct = C0t ∪ C1

t for allt 6= 0 as in the following figure 3.1

Figure 3.1: Are there limits that do notcreate embedded points?

Note that in level of ideal sheaves: the flat limit of ICt as t→ 0 is notthe same as IC1

0∪C01. What about limt→0 OCt ? This one is the same as


limt→0 OC0t⊕OC1

t, in other words the sheave OC1

0⊕OC0

0is a flat limit of

OCt . One can conclude that maybe we need to work with these typesof sheaves rather than ideal sheaves.

The first issue with this, is that flat limits are not unique. There arenon-trivial extensions

0→ OC10→ F → OC0

0→ 0

which occur as flat limits. This tells us that the moduli space is notseparated.

The second issue is that the extensions of OC00

by OC10

have morecomplicated automorphism groups. Sheaves like OC0

t⊕OC1

thave auto-

morphism groups which can vary in moduli. The upshot of this is thatthe moduli space of such sheaves is a (non-separated) Artin stack.

We need to “remember” that OCt is a structure sheave; we have apair consisting of a sheaf and a surjective map f from OX :

f : OX → OCt .

But f can have limits that are not surjective! And the definition ofstable pairs due to Pandharipande and Thomas is the right way todream a moduli space for these objects that is proper (the limits exists)and is separated (the limits are unique).Definition 3.1 (Pandharipande-Thomas). Let X be a threefold. A sta-ble pair (F , s) is a sheaf F of dimension 1 and a map OX

s−→ F (hencea section s ∈ H0(X,F )) such that1. F is pure (i.e. there does not exist any non-zero G → F such that

dimG = 0 ); and,2. The coker(s) is dimension zero (so s cannot vanish on a curve).

The first condition eliminates possibility of existence of embeddedpoints in case of curves for example (this is equivalent to the sheafbeing Cohen-Macauly in case of curves). The second condition aboveis as much as we want s to be close to a surjection. The natural notionof equivalence of pairs is that, two pairs s : OX → F and s : OX → F ′are equivalent if there exists an isomorphism F ∼= F ′ making thefollowing triangle commute:

OXs //

s′

F∼=��

F ′

In this case our discrete invariants are n = χ(F ) and note thatker(s) = IC, for [C] = β ∈ H2(X). The above pairs had been previouslystudied, and the main theorem that tells us this is the correct definitionis the following

pandharipande-thomas theory 51

Theorem 3.2 (LePotier, Pandharipande-Thomas). The moduli space ofstable pairs with fixed n, β is a proper and separated schemes:

PTn(X, β) = { stable pairs OXs−→ F ; n = χ(F ), β = [suppF ]}.

In the example we have

0→ IC0t ∪C1

t→ [OX

s−→ OC0t⊕OC1

t]→ 0

when t 6= 0 and the limit is

0→ IC00∪C1

0→ [OX

s−→ OC00⊕OC1

0]→ Op → 0.

So the cokernel of the limit stable pair, is not flat but is a skyscrapersheaf.

Let OX → F be a stable pair with supp(F ) = C a smooth curve. SoF must be a line bundle on C with a non-trivial section; i.e. F = O(D)

for D = ∑ ni pi an effective divisor (figure 3.2).

Figure 3.2: Points (given by divisor D)are now constrained to lie on the curve(C)

So we still have points and curves but the points are now con-strained to lie on the curve. If C ⊂ X is an isolated smooth curveof genus g then

PTn(X, [C]) = Symn+g−1(C)

where n = 1− g+ |D|. Comparing this to In(X, β) we see that PTn(X, β)

is typically smaller and nicer. We will see that the PT moduli space isthe most efficient moduli space as far as curve counting is considered.

Recall that the bounded derived category Db(Coh(X)) is the cate-gory of bounded complexes

E• = {· · · → E−1 → E0 → E1 → · · · }

where we localize the quasi-isomorphisms. The deformation theoryof objects in this category is expressed in terms of trace free Hom,Hom0(E•, F•).Theorem 3.3 (Thomas). If we consider deformations of pairs O

s−→ F as ob-jects {I•} of the derived category then the moduli space is the same PTn(X, β)

but the obstructions change. The space of deformations stays the same:

Def({I•}) = Ext10(I•, I•)

but the obstructions change

Def({I•}) = Ext20(I•, I•).

Feb 28

A conclusion is that PTn(X, β) is locally given by

κ = d f : Ext10(I•, I•)→ Ext2

0(I•, I•) ∼= Ext11(I•, I•)∨


where f : Ext10(I•, I•) → C. Our previous machinery goes through

and we have

NPTn,β(X) =

∫[PTn(X,β)]vir

1 = evir(PTn(X, β))

where the right hand side is the Euler characteristic weighted by Behrendfunction. We have the analogous definitions

ZPTβ (X) = ∑

nPTn,β(X)qn

ZPT(X) = ∑β

ZPDβ (X)vβ.

Note that from the definitions if β = 0 the only stable pair OX0−→ 0 and

ZPT0 (X) = 0.

Example 3.4. If C ⊂ X is a smooth curve of genus g, a stable pairsupported in C must be OX

s−→ O(D) where D = ∑ ni pi is an effectivedivisor (ni > 0), and n = χ(O(D)) = deg(D) + 1− g. Pairs supportedon C of euler characteristic n give the moduli space Symn+g−1(C).Example 3.5. Let X = tot(O(−1)⊕O(−1)→ P1) be the local P1. Then

PTn(X, [P1]) = Symn−1(P1) = Pn−1

and NPTn,[P1]

(X) = (−1)n−1n. The partition function is

ZPT[P1](X) =

∞

∑n=1

(−1)n−1nqn

= −∞

∑n=1

n(−q)n = − −q(1− (−q))2

=q

(1 + q)2 .

Which we recall to be the same as ZPT[P1]

(X)′.Now the goal of this theory is achieved by the conjecture of Pandharipande-

Thomas which was proved first by Todo and later by Bridgeland usingmachinery of Joyce-Song and Kontsevich-Soiblemann:Theorem 3.6 (Todo, Bridgeland).

ZPTβ (X) =

ZDTβ (X)

ZDT0 (X)

.

Example 3.7. Let N → C be a rank 2 bundle on a smooth curve ofgenus g with ∧2N ∼= KC and such that H0(C, N) = 0 and consider thelocal curve X = tot(N). Then the zero section C ↪→ N is rigid and

PTn(X, [C]) = {OXs−→ L, L = O(D), |D| = n + g− 1} = Symn+g−1(C)

which is smooth of dimension n + g− 1. And

NPTn,[C](X) = (−1)n+g−1e(Symn+g−1(C)).

pandharipande-thomas theory 53

Then there is a theorem due to MacDonald in topology that for anytopological space Y, we have

∞

∑k=0

e(Symk(Y))yk =1

(1− y)e(Y).

So we can compute

ZPT[C](X) =

∞

∑n=1−g

(−1)n+g−1e(Symn+g−1(C))qn

= q1−g∞

∑k=0

e(Symk(C))(−q)k = q1−g(1 + q)2g−2

= (q−1/2 + q1/2)2g−2 = (q−1+ 2 + q)g−1.

We may now check the correspondence here by change of variableq = −eiλ:

ZPT[C](X) = (2− e−iλ − eiλ)g−1

= (2− 2 cos(λ))g−1 = (4 sin2(λ

2))g−1 = (2 sin(

λ

2))2g−2.

So

ZGW[C] (X)′ = (λ− 2

6λ3

8+ · · · )2g−2

= λ2g−2(1− λ2

24+ · · · )2g−2

= 1.λ2g−2 − g− 112

λ2g + · · · .

Recall that these invariants a priori count the possibility disconnectedGromov-Witten invariants. But because of geometry of C there are nomaps of disconnected curves, hence these are actually the connectedGromov-Witten invariants:

ZGW[C] (X)′ = FGW

[C] (X)′.

And the Gupakumar-Vafa conjecture is saying that

FGW(X)′ = ∑β=0,g≥0


= ∑β 6=0,n≥0

nn,β(X)∞

∑d=1

1d(2 sin(

dλ

2))2g−2vdβ.

And FGW[C] (X)′ is the v[C] term of above. So

nh,[C] =

{1 h = g0 h 6= g.

But what are nh,d[C]s? We yet have to answer!

Chapter 4Gupakumar-Vafa Invariants

Mar 3

So far we have been studying three cornerstones of the general picture4.1. Physics interpretation suggests that Gupakumar-Vafa invariantsmust be the most fundamental invariants among the ones we are in-vestigation. But the issue with them is that unlike the other invariantsit is hard to invent a rigorous mathematical framework for them. Oneidea is to try to use the conjectural correspondences between otherinvariants and potential Gupakumar-Vafa invariants as means of defi-nition of them.

Figure 4.1: Curve-counting invariantsand their correspondences

4.1 BPS conjecture

Recall that

ZPT = ∑β

ZPTβ (q)vβ

= ∑β

ZGWβ (λ)′vβ

= exp

(∑β 6=0

FGWβ (λ)vβ

)

= exp

(∑β 6=0

∑g≥0


)


which results in view of the multiple-cover formula

ZPT = exp

(∑

β 6=0,g≥0ng,β(X) ∑

d>0

1d

(2 sin

(dλ

2

))2g−2vdβ

)

= exp

(∑

g≥0,β 6=0ng,β ∑

d>0

1d

(1i

(e

idλ2 − e

−idλ2

))2g−2vdβ

)

= exp

(∑

g≥0,β 6=0ng,β ∑

d>0

1d

(−e−idλ(eidλ − 1)2

)g−1vdβ

)

= exp

(∑

g≥0,β 6=0ng,β ∑

d>0

(−1)g−1

d

((−q)−d(1− (−q)d)2

)g−1vdβ

)

So we have the following formal identity between these invariants

∑β,n

NPTn,β(X)qnvβ =

exp

(∑

g≥0,β 6=0ng,β ∑

d>0

(−1)g−1

d

((−q)−d(1− (−q)d)2

)g−1vdβ

)(4.1)

So one can try to take this as the definition of Gupakumar-Vafainvariant. In fact, any series of the form of the left hand side, where thecoefficients of vβ are Laurent series in q with integer coefficients, can bewritten in the form of the right hand side if we sum g = −∞, · · · , c(β)

with ng,β ∈ Z.So the upshot of this story is that we can use 4.1 with allowing g to

be negative as well, to define ng,β(X) in terms of NPTg,β . Then ng,β ∈ Z

is automatic. But the conjecture now is that

BPS Conjectureng,β = 0, ∀g < 0 (4.2)

A curve class β ∈ H2(X) is is said to be irreduicble if β 6= β1 + β2

with βi’s being effective.We can look at vβ in for irreducible classes (hence vβ 6= vβ1 vβ2 ) so

ZPTβ (q) = ∑

gng,β(−1)g−1

(− (1 + q)2

q

)g−1

= ∑g

ng,βq1−g(1 + q)2g−2

= ∑g

ng,β(q−1+ 2 + q)g−2

Any Laurent series in q can be written in the form ∑r≤C arq1−r(1 +q)2r−2 since

q1−2(q + 1)2r−2 = q1−2 + higher order terms .

gupakumar-vafa invariants 57

We can prove the BPS-conjecture in this case (for irreducible β). Wewish to show that for an irreducible class β we have

ZPTβ = ∑

rNPT

n,βqn =C(β)

∑g=0

ng,βq1−g(1 + q)2g−2.

For g ≥ 1 the Laurent polynomials

q1−g(1 + q)2g−2 = (q−1+ 2 + q)g−1

are 1, q−1+ 2 + q, q−2 + 4q−1, 6 + 4q + q2, · · · .For g = 0, we have

1(1 + q)2 = q− 2q2 + eq3 + · · · = ∑(−q)n−1nqn.

So we want to prove that

NPTn,β = NPT

−n,β + c(−1)n−1n, c = n0,β

We will see that the proof is a consequence of Serre duality. Notethat when we prove this we also prove a part of the MNOP conjecturewhich is that we are allowed to make that substitution.

Let Mn(X, β) be the moduli space of stable, pure sheaves F withχ(F ) = n and such that [supp(F )] = β. This space is a scheme byresults of Simpson.

Let β be irreducible, then if Os−→ F is a PT-pair then F is stable.

The mapΦn : PTn(X, β)→ Mn(X, β)

mapping (F , s) 7→ F is therefore well-defined. And the fibers of thismap are Φ−1

n ([F ]) = P(H0(X,F )). We stratify Mn(X, β) by the di-mension of H0(X,F ):

Mn(X, β) =⊔α

Vα.

Then Φ−1n(Vα)→ Vα is a fiber bundle with fibers Pα−1so e(PTn(X, β)) =

∑α αe(Vα). So the Behrend function is compatible with Φn.Lemma 4.1 (PT). We have νPTn(X,β) = (−1)n−1Φ∗n(νMn(X,β)). Thus wewill have

NPTn,β(X) = evir(PTn(X, η)) = ∑

α

α(−1)n−1evir(Vα).

There is an isomorphism Mn(X, β)∼=−→ M−n(X, β) via F 7→ F∨.

PTn(X, β)

Φn��

PT−n(X, β)

Φ−n��

Mn(X, β) //M−n(X, β)


with fibers Φ−1−n(Vα)→ Vα being isomorphic to Pα−n−1.Also note that

H0(X,F∨) = H0(X, R•Hom(F , O))

where a priori R•Hom(F , O) is a complex of sheaves with cohomologyExti(F , O). But because codim(suppF ) = 2 we have Exti<2 = 0 andsince F is pure we have Ext3 = 0. Therefore

H0(X,F∨) = H0(R, Ext2(F , O)) = Ext2(F , O) = H1(X,F )∨

by Serre duality on Calabi-Yau threefolds.Of course n = h0(F )− h1(F ) = h0(F )− h0(F∨). . . . So we have

evir(PT−n(X, β)) = ∑α

(α− n)(−1)n−1evir(Vα)

and hence

NPTn,β − NPT

−n,β = ∑α

α(−1)n−1evir(Vα)− (α− n)(−1)−n+1evir(Vα)

= (−1)n−1n ∑α

evirVα = (−1)n−1nevir(Mn(X, β)).

Finally evir(Mn(X, β) = evir(Mn+1(X, β)). This is true because locallythe two spaces Mn(X, β) and Mn+1(X, β) are isomorphic via [F ] 7→[F (D)] for some D.β = 1. But such line bundle D exists locally bysome technical details.Conjecture 4.2 (Katz). For any β,

n0,β(X) = evir(M1(X, β))

where the right hand moduli, is the moduli space of stable sheaves with eulercharacteristic one.

Chapter 5GeneralizedDonaldson-ThomasInvariants

Mar 5

Our motivation for embarking on this topic is Bridgeland’s proof ofDT/PT correspondence through techniques of Wall-crossing and Hallalgebras and also Li-Kiem’s geometric definition of Gupakumar-Vafainvariants. Recall that we resorted to defining GV-invariants usingother invariants and saw that finally in genus zero we can get a defi-nition using Donaldson-Thomas type invariants, which is a shame be-cause somehow Gupakumar-Vafa invariants are supposedly the mostfundamental of our invariants; one’s that are closest to actual curve-counting.

What are the basic ingredients for the construction of Donaldson-Thomas (Pandharipande-Thomas) invariants:1. M is some moduli space of coherent sheaves (with objects in Db(Coh(X)))

on a Calabi-Yau threefold X, andM is a finite type scheme.2. Deformation theory is such that the moduli space locally at [E] ∈ M

is modelled on {d f = 0} ⊂ Def(E) where the potential function isf : Def(E)→ C.

3. Behrend function νM :M→ Z defined via

νM([E]) = (−1)dim Def(E)(1− e(MFf (E))).

4. Numerical invairants:

evir(M) := ∑k∈Z

ke(ν−1M(k)).


What did we really use here to get (1)–(4)? Property (2) is a cat-egorical property of Coh(X), or more generally that of Db(Coh(X)).So suppose that C is an abelian category with a notion of families ofobjects in C (for C = Coh(X) this is flat families). The infinitesimaldeformations of E ∈ Ob(C) are classified by Ext1(E, E) and the ob-structions are given by Ext2(E, E) and the moduli stack is locally givenby

Ext1(E, E) κ−→ Ext2(E, E)

and the moduli stack of objects in this category is locally expressed via{κ = 0}/ Aut(E). If Ext1(E, E) ∼= Ext2(E, E)∨ then κ = d f .Definition 5.1. An abelian category C is called Calabi-Yau of dimen-sion n if

Exti(E, F) ∼= Extn−i(F, E)∨

for all objects E and F in C, and the moduli stack of objects in C formsan algebraic stack with stabilizers which are affine algebraic groups.

Any CY3 category satisfies property (2). For purposes of our course,Remark 5.2. Recall that such stacks are locally quotient stacks [X/G]

with X a scheme and G is an affine algebraic group actin on X. In thestack world the quotient is “free” even if the action is not. If X is amanifold and G is finite the [X/G] is an orbifold so we may think ofthese as generalization of orbifolds.

A theory of Donaldson-Thomas invariants for CY3 categories waspredominantly invented by Kontsevich-Soibelmann and Joyce-Song.There are mainly two examples in which this goes through:1. C = Coh(X) as the abelian category of coherent sheaves on a Calabi-

Yau threefold (that of Joyce-Song) and Db(Coh(X)) specifically inframework of Kontsevich-Soiblemann;

2. C = Rep(Q, w), the category of representations of a quiver Q witha superpotential w .

3. C the flat complex-bundles on real oriented 3-manifold.There are some simple categories that can be interpreted as all threeabove. For example if C = Coh(C3) is the category of sheaves withcompact support on C3, then this category is also isomorphic toRep(Q, xyz−xzy) where Q is

• xvv

y

��

z66

Also C is the category of flat bundles on T3 = S1× S1× S1, or equiva-lently the category of C[x, y, z]-module or that or triples of commutingmatrices. Recall that in this case the superpotential is given via

f : (Mn×n(C))3 → C

mapping (X, Y, Z) → tr(X[Y, Z]). The condition of d f = 0 is equiva-lent to [X, Y] = [Y, Z] = [X, Z] = 0.

generalized donaldson-thomas invariants 61

Recall that in ordinary (curve counting) Donaldson-Thomas theorywe fixed numerical invariants n, β. We index components of the mod-uli space M, of objects in C by α ∈ K(C). So Mα is the moduli space ofobjects E with [E] = α ∈ K(C). Mα typically fails to abe a scheme, it isa stack locally of the form [{d f = 0}/ Aut E] where f : Ext1(E, E)→ C.

We can still define νMα . Basic property of Behrend function: If

F // P

��

E

where π is smooth then

νp = π∗(νB).(−1)dim F

is the family version of the fact νsmooth = (−1)dim X . In particular if Gacts freely on X then

G // X

π��

X/G

and νX = (−1)dim Gπ∗(X/G) for a stack quotient [X/G]. So forwe define ν[X/G] so that the above holds.. But then property (4)

above is not satisfied: For a Zariski locally trivial bundle

F // p

��

B

we must have e(P) = e(F).e(B) and evir(P) = evir(F).evir(B).If we want to define evir[X/G] we are forced to have evir[X/G] =

evir(X)evir(G)

where evir(G) is zero unless dim G = 0. So we can do one of thetwo things:• We need to restrict to open substacks M◦α ⊂ Mαwhich is an actual

scheme (this involved choosing s stability condition); or,• We could generalize our notion of Euler characteristic to take values

in a ring where the generalized euler characteristic of a group G isinvertible in the ring K0(VarC) the Grothendieck group of varieties(a.k.a. ring of motives or ring of motivic classes).

5.1 Grothendieck groups

Definition 5.3. The Grothendick group of varieties, K0(VarC), is defindto be the free abelian group generated by isomorphism classes of vari-eties with the scissor relation

[V] = [V r Z] + [Z]


where Z ⊆ V is a closed subvariety.K0(VarC) is a ring under the cartesian product: [V].[W] = [V ×W].

Immediately we see that if

F // P

��

B

is a Zariski locally trivial fibration then [P] = [F].[B]. Note that vari-eties are by definition of finite type (which is needed for noetherianinduction for proof of this fact).

A few distinguished elements of this ring are [∅] = 0 which is theadditive identity, the motivic class of a point [pt] = 1 which is themultiplicative identity. The class of affine line is known as the Lefschetzmotive L = [C]. It is easy to see that

[Pn] = Ln + Ln−1 + · · ·+ 1.

There is a ring homomorphism e : K0(Var/C) → Z via [V] 7→ e(V).Note that e(L) = 1. In fact there are more general ring homomor-phisms as such:

pt(.) : K0(Var/C)→ Z[t]

via [V] 7→ pt(V) such that if V is a projective manifold then pt(V) =

∑2 dim Vk=0 dim Hk(V, Q)tk is the Poincaré polynomial of V. And note that

pt(L) = t2. In world of varieties, pt is called the Serré polynomial orsometimes “virtual Poincaré polynomial”.Example 5.4. Let X = BlC(P3) where C is a curve of genus g. Then[X] = [P3 rC] + [E] where E is the exceptional divisor P(NC/P3)→ C.One can check that the normal bundle NC/P3 is Zariski locally trivialon C and hence so is the projectivization. The fibers of E → C areisomorphic to P1 so we get

[X] = [P3]− [C] + [P1].[C] = 1 + L + L2 + L3 + L.[C].

The Poincaré polynomial is hence

pt(X) = 1 + t2 + t4 + t6 + t2(1 + 2gt + g2)

= 1 + 2t2 + 2gt3 + 2t4 + t6.

The coefficients are therefore the Betti numbers h0 = h6 = 1, h2 = h4 =

2 and h3 = 2g.Another easy computation is the motivic class of the general linear

group:Example 5.5. We have a fibration Gln(C) → Cn r {0} by mappingany matrix to its first column vector. The fiber of this map over vector


v1, called F1v1

has a Zariski locally trivial on 2nd column F1v1→ Cn r

span{v1} = Cn r C and we can keep going inductively to get

[GLn(C)] = L(n2)(Ln − 1)(Ln−1 − 1) · · · (L− 1)

and by notation L(n2)[n!]L.

Note that GLn is not a projective variety, and in fact the virtualPoincaré polynomial of it has many roots in 1. But there is a nice wayof making projective varieties from them:Example 5.6. Let Gr(k, n) be the Grassmanian of k planes in Cn. Wehave

[Gr(k, n)] =[GLn]

[GLk][GLn−k][Ck(n−k)]

=L(n

2)[n!]L

L(k2)[k!]LL(n−k

2 )[(n− k)!]LLk(n−k)

=[n!]L

[k!]L[(n− k)!]L

because (n2) = (k

2) + (n−k2 ) + k(n− k). By notation this is written as(

nk

)L

=(Ln − 1)(Ln−1 − 1) · · · (Ln−k+1 − 1)

(Lk − 1) · · · (L− 1)

Note that by substituting L with 1, we get the usual binomial polyno-mial (n

k) though this can only be done through residues. The Poincarépolynomial is

pt(Gr(k, n)) =(t2n − 1)(t2n−2 − 1) · · · (t2n−2k+2 − 1)

(t2k − 1) · · · (t2 − 1).

So for instance [Gr(2, 4)] = 1 + L + 2L2 + L3 + L4 and hence the Bettinumbers are respectively 1, 0, 1, 0, 2, 0, 1, 0 and 1.

If we look at K0(VarZ) ⊂ K0(VarC) then we get homomorphisms bycounting Fq-points of the varieties:

#Fq : K0(VarZ)→ Z.

This gives another way of finding Poincaré polynomial of varieties.Another ring homomorphism from K0(Var/C)→ Z is the Serre poly-nomial, which in particular maps L to t2. In next class we will defineGrothendieck groups of other categories of spaces of our interest.

Mar 10

Let K0(Sch/C) be the free abelian group generated by isomorphismclasses of schemes of finite type over C with scissor relations. AlsoK0(St/C) is the free abelian group of isomorphism classes of stacks offinite type over C with their analogous scissor relations. We have

K0(Var/C) ↪→ K0(Sch/C) ↪→ K0(St/C).


Lemma 5.7. K0(Var/C) is isomorphic to K0(Sch/C).The idea is that if for instance we consider X = Spec C[x]/(x2), then

Xred is a point. Let

[X] = [A1]− [A1 r X] = [A1]− [A1 r Xred = [Xred]

because the Grothendieck group can only see the topology of the un-derlying space.1 1 This is because of Bridgeland’s rela-

tions that identifies [X] and [Y] if thereis a geometric bijection f : X → Y

between them. Otherwise it is nottrue only under scissor relations.

5.1.1 Grothendieck group of stacks

We will assume that all stacks are locally of finite type and have affinestabilizers. In particular this implies that we can locally write them asquotient stacks [X/G] for X a scheme of finite type, and G ⊂ Gln(C)

an affine algebraic group.Theorem 5.8. We have isomorphisms

K0(St/C) ∼= K0(Var/C)[GLn(C) : n = 1, 2, · · · ]∼= K0(Var/C)[L−1, (1−Ln)−1 : n = 1, 2, · · · ]

The idea of the proof is to find a stratification of X such that eachstrata is of the form [Y/G] where Y is a scheme and G ⊂ Gln(C). Thenwe have

[Y/G] = [(Y×GLn /G)/ Gln] =[Y×GLn /G]

[Gln].

Also note that K0(Var/C)[L−1, (1−Ln)−1 : n = 1, 2, · · · ] is a subringof the completion of K0(Var/C) in L-adic topology:

K0(Var/C)[L−1, (1−Ln)−1 : n = 1, 2, · · · ] ⊂ K0(Var/C)[[L−1]].

5.1.2 Relative Grothendieck groups and relative motives

Recall that our goal is to make sense of evir(M) and νM such thatitems (3) and (4) above hold. We need a motivic version of Milnorfiber for this matter, i.e. one that is defined in terms of motives ratherthan topological spaces. And in place of a constructible function likeν and a weighted euler characteristic with respect to it, we need afunction valued in motives and a motivic version of the weighted eulercharacteristic.

The second part is addressed by relative motives. Let S be a varietyover C. Then K0(Var/S) is defined as the free abelian group generatedby S-varieties [ f : X → S], with relation

[ f : X → S] = [ f |U : U → S] + [ f |Z : Z → S]

where Z ⊂ X is closed subvariety and U = X r Z is the complement.Similarly one can defined the relative Grothendieck group of stacksK0(St/S) where S can be a stack.


Given ϕ : S→ T a map of stacks, we can defined the pushforward

ϕ∗ : K0(St/S)→ K0(St/T), via[ f : X→ S] 7→ [ϕ ◦ f : X→ T]

and pullback

ϕ∗ : K0(St/T)→ K0(St/S), via[ f : X→ T] 7→ [X×T S→ T].

So we can think of [X → S] ∈ K0(Var/C) as a motive-valued func-tion on S which to each S-point i : S0 ↪→ S assigns

i∗([X → S]) ∈ K0(Var/S0) = K0(Var)

we may then compose this with the function K0(Var/S)→ K0(Var/C)

via [X → S] 7→ [X] to get a weighted euler characteristic: more specif-ically we get a constructible function

ν : S→ Z viaS0 7→ e( f−1(S0)).

Then

e(S, ν) = ∑k∈Z

ke(ν−1(k)) = e([X]).

Mar 12

5.2 Hall algebra of stacks

Let C be a CY3 category. Our favorite examples are C = Coh(X)

or the stack of cherent sheaves of dimension ≥ 1 on a Calabi-Yauthreefold, C = Coh≤1(X). The Hall-algebra, H(C), is additively theGrothendieck group, K0(St/C), of stacks relative to C (i.e. by abuseof notation, the moduli stack of objects in C). Recall that K0(St/C)consists of classes

[ f : X→ C]

of stacks of finite type X with the scissor relations

[ f : X→ C] = [ f |Z : Z→ C] + [ f |XrZ : Xr Z→ C].

The Hall product

[X→ C] ∗ [Y→ C] = [Z→ C]

is the C-stack Z, of triples (x, y, E) where x and y are objects in X andY and E is an extension

0→ f (x)→ E→ g(y)→ 0.


More categorically, let Cext be the stack of short exact sequences in C.And we define maps to C via

pi : (0→ E1 → E2 → E3 → 0) 7→ Ei.

and the Hall product ∗ : H(C)⊗H(C)→ H(C) is defined as (p2)∗(p1×p3)∗ as in the following diagram

Z //

��

Cext

��

// C

X×Yf×g// C× C

Remark 5.9. The multiplicative unit in H(C) is 1 = [0→ C].

5.2.1 Associativity

Let C f lag be the stack of flags F1 ⊂ F2 ⊂ F3 of objects Fi in C. Then thefollowing diagram is easily seen to be cartesian:

C f lag

�

//

��

Cext

��

Cext × C // C× C.

Here the upper horizontal map is

(F1 ⊂ F2 ⊂ F3) 7→ (0→ F2 → F3 → F3/F2 → 0)

and the left vertical map is

(F1 ⊂ F2 ⊂ F3) 7→ (F1 → F2 → F2/F1, F3/F2).

Then the composition H(C)⊗H(C)⊗H(C)∗⊗id−−→ H(C)⊗H(C) −→

∗H(C) is induced by the diagram

C f lag

�

//

��

Cext //

��

C

Cext × C //

��

C× C

C× CC

So ∗ ◦ (∗ ⊗ id) : H(C)⊗ H(C)⊗ H(C) → H(C) is induced actuallyby

C f lag

��

// C

C× C× C


where the vertical map is F1 ⊂ F2 ⊂ F3 7→ (F1, F2/F1, F3/F2) and thehorizontal arrow maps the same flag to F3.

The observation is that the other composition ∗ ◦ (id⊗∗) is alsoinduced by the same diagram. This proves the associativity. Alsonotice that this gives a means of thinking about n-fold Hall products.

5.2.2 Equations in Hall algebras

Fix C = Coh≤1(X) for a Calabi-Yau threefold X. Let P ⊂ C be thesubcategory of P = Coh0(X) of sheaves of dimension zero. Let Q ⊂ C

be the subcategory of pure sheaves of dimension one, i.e. for any objetQ in Q we have Hom(P, Q) = 0 for all objects P of P. In this case wesay Q = P⊥.

Note that OXs−→ F is a PT stable pair if and only if F is an object of

Q and coker(s) is an object of P. The pair (P, Q) is called a torsionpair; that is to say any E ∈ C can be written uniquely as an extension

0→ P→ E→ Q→ 0

for some P and Q respectively objects of P and Q.We are interested in the elements

1X = [X ↪→ C]

where X is the moduli stack of objects in X for cases of substacks P, Qand C of C. Note that strictly speaking these elements are not in H(C)

because they are not of finite type. There are two ways of overcomingthis:1. We can grade H(C) by elements λ in K-theory of X, K0(X) and then

1X is finite type over each component Cλ. The Hall product will bedefined over ⊕λ Hλ and respects the grading;

2. We may think of these as formal sums of elements in H(C), in fact

as elements in a formal completion H(C).This gives an identity in the Hall algebras:

1C = 1P ∗ 1Q. (5.1)

Another element we are interested in is

I = [I•(X, •)→ C]

where I•(X, •) = ⊔n,β In(X, β) and maps (OX → C) 7→ C. Let C(O) be

the stack of pairs (F, s) where F is an object in C and s : OX → F. ThenI•(X, •) is an open substack of C(O). Let 1O

C be the element [C(O)→ C]

where the structure morphism is [OX → E] 7→ E.The following identity

1OC = I ∗ 1C (5.2)


says that every map factors uniquely as a surjection followed by aninjection.

Given OXf−→ F in C(O) we can uniquely construct a diagram

OX

zzzz

f��

0 // OX/ ker F // F // E // 0

representing an element in the stack I ∗ 1C.Conversely given an element in I ∗ 1C, i.e.

OX

��

0 // OC // F // E // E/F // 0

we get OXf−→ F by composition .

Notice that our Hall algebra is an algebra over K0(Var/C) via theproduct

[S].[ f : X→ C] = [X× S→ C].

Also sinceK0(St/C) = K0(Var/C)[L−1, (1−Ln)−1]

our Hall algebra is a K0(Var/C)[L−1]-algebra via the induced struc-ture.

Let Hreg(C) ⊂ H(C) be the sub K0(Var/C)[L−1]-module generatedby [X→ C] where X is a scheme.Lemma 5.10. Hreg(C) ⊂ H(C) is a K0(Var/C)[L−1]-subalgebra.Proof. The content of proof is to show that when X and Y are stacks,the product [X → C] ∗ [Y → C] can be written in terms of schemes overC. Let [Z → C] denote this Hall product. Then Z is the stack of triples(x, y, E) for x ∈ X and y ∈ Y and extension

0→ f (x)→ E→ g(y)→ 0.

So Z is locally written as Z/Cn where Z is a scheme. In fact thestructure of Aut(x, y, E) is Aut x×Aut y o Hom(g(y), f (x)):

Aut(x, y, E) =

(Aut x Hom(g(y), f (x))

0 Aut y

).

But when X and Y are schemes Aut x and Aut y are trivial.

5.3 Integration map

A naive integration map is

Φeul : Hreg(C)→ Z[K(C)]


via [ f : X → C] mapping to ∑α∈K(C) e(Xα)qα where Xα = f−1(Cα) forall elements in K-theory, α = (0, 0, β, n) and using notation qα = qnvβ

notice that this is close to our DT partition functions.A better integration map will incorporate virtual euler characteris-

tics:Φ : Hreg(C)→ Z[K(C)]

via [ f : X → C] mapping to ∑α∈K(C) e(Xα, f ∗νC)qα

It turns out that for maps f : In(X, β) → C the pullback of Behrendfunction is

f ∗νC = (−1)nνIn(X,β).

So for I = [I•(X, •)→ C] we have

Φ(I) = ∑n,β

(−1)nevir(In(X, β))qnvβ = ZDt(−q, v).

Let analogously define PT = [PT•(X, •) → C with structure map[OX → F] 7→ F. Not that PT also has same type of stratification bycurve class and euler characteristic, PT•(X, •) = tn,βPTn(X, β).Lemma 5.11. We have another Hall algebra identity

1CQ = PT ∗ 1Q (5.3)

Proof. Given Of−→ E and object in Q(O) we can construct the following

diagram0 // OX

��

// OX

f��

// 0

��

0 // F //

��

E

��

γ// Q //

��

0

0 // P //

��

coker f //

��

Q //

��

0

0 0 0

We can complete the diagram as shown by diagram chasing. Noticethat F = ker γ so OX

s−→ F is a PT pair. Also since Q is extension free Fis in Q because E and Q are. So P is in P proving one direction. Theother direction of the claim follows analogously.Lemma 5.12. Yet another identity in the Hall algebraProof. The right hand side is stack of objects

OX

��

OX

��

0 // P // E // Q // 0


and C(O) is the stack of objects

OX

��

0 // P // E // Q // 0

where both are Zariski locally trivial fibrations over the stack of objects

OX

��

0 // P // E // Q // 0

The fiber of the first one is H0(P) and the fiber of second map isker(H0(E)→ H0(Q)) But H1(P) = 0 in the long exact sequence

0→ H0(P)→ H0(E)→ H0(Q)→ H1(P)

because P is supported on points. This is enough to prove the identityin the Hall algebra.

5.4 Bridgeland’s argument

Starting from identity 5.2 and using identities 5.1 and 5.3 we get

I ∗ 1P ∗ 1Q = I0 ∗ 1P ∗ PT ∗ 1Q. (5.4)

If we could prove that 1Q and 1P were invertible in Hall algebra. Thenwe would get

I = I0 ∗ 1P ∗ PT ∗ 1−1P .

This is true and needs to be proven.Now let’s pretend we can apply Φ (which we cannot the way we

have defined it). Then we get

Φ(I) = Φ(I0)Φ(1P)Φ(PT).Φ(1P)−1= Φ(I0)Φ(PT)

which translates to

ZDT(−q, v) = ZDT(−q, 0).ZPT(−q, v).

There are two approaches to defining Φ(1P):1. Joyce-Song: Use stability to show that 1P = exp

(ε

L−1)

for ε ∈ H(C)

a regular element. Then show that Φ is well-defined for 1P ∗ PT ∗1−1P .

2. Kontsevich-Soiblemann: enlarge Z[K(C)] to something over whichwe can take Φ of stacks (leading to motivic DT/PT correspondence).


Mar 17

Recall that for any Calabi-Yau category C of cohomological dimen-sion ≤ 1, we derived an equation in the Hall algebra H(C) (i.e. equa-tion 5.4) in terms of classes C-stacks of moduli of objects of the follow-ing subcategories:• I0, ideal sheave of 0 dimensional subschemes;• PT of PT pairs;• I, of ideal sheaves of 1 dimensional subschemes;• P, all sheaves of dimension 0;• Q, pure sheaves of dimension 1.

But in order for our DT/PT program to go through we need toextend the integration map

Φ : Hreg(C)→ Z[[K(C)]]

in a way that integration makes sense in bigger subalgebra of H(C).In fact we will attempt to define Φ on all of H(C). We need to enlargethe coefficient ring of range of Φ to some R[[K(C)]]. In fact as we willsee R will be a sopped up version of K0(St/C) = K0(Var/C)[L−1, (1−Lk)−1 : k ≥ 1].

Recall that if E is an object of C and C is locally near E givenby {d f = 0}/ Aut(E) where f : Ext1(E, E) → C then νC(E) can beexpressed as

νC(E) = ν{dt=0}/νAut(E) = (−1)−dim Ext0(E,E)ν{d f=0}(E)

in fact such identity is true for Behrend functions even if this quotientis not Zariski locally trivial. We can write this now as

νC(E) = (−1)−dim Ext0(E,E)+dim Ext1(E,E)(1− e(MFf (0))).

We want as assignment E 7→ w(E) ∈ Mot (where Mot is the ring ofmotivic measures) such that e(W(E)) = νC(E). Here we have

Mot = Kµ0 (St/C)[L

1/2] = Kµ0 (Var/C)[L−1, (1−Ln)−1, L

1/2]n≥1.

For w ∈ Mot we can define euler characteristic of e(w) (could be ∞) byassigning e(L1/2) = −1. The motivic version of Milnor fiber appearsin the definition

w(E) := (L1/2)−dim Ext0(E,E)+dim Ext1(E,E)(1−MFf ),

where MFf here is the motivic Milnor fiber.With some work we will see that E 7→ w(E) comes form an element

W → C] ∈ K0(St/C)[L1/2], such that by pulling back along i : {E} ↪→C, we get

i∗[W → C] = w(E).


So we need some sort of compatibility that guarantees this. We thendefine

Φmot([X → C]) := ∑α

[Xα ×C W]qα ∈ Mot [[K(C)]].

And all this happens to work because of the followingTheorem 5.13 (Kontsevich-Soiblemann). Φmot : H(C) → Mot [[K(C)]]is a homomorphism through which Φ : Hreg → Z[[K(C)]] factors.

5.5 Motivic Milnor fiber

Let f : X → C be an analytic or algebraic functino on smooth varietyX of dimension d. Assume that crit( f ) = {d f = 0} ⊂ X0.Definition 5.14. Let x0 ∈ X0. Then the Milnor fiber of f at x0 is definedas MFf (x0) = f−1(δ) ∩ Bε(x0) when 0 < δ� ε� 1.

Note that f−1({|y| = δ}) is a fibration over S1 with fibers diffeomor-phic to MFf (x0). This provides us with a monodromy action on theMilnor fiber. The cohomology H∗(MFf (x0)) now is equipped with amonodromy action (of finite order) and so we can view it as an actionof µN = {zN = 1} ⊂ C× on H∗(MFf , Z) for some N. The jargon isthat e(MFf (x0)) is called the Milnor number of f and the elements ofH∗(MFf , Z) are called nearby cycles.

Let µ = proj lim µn. We define K0(µ)(Var/X0) as the free abeliangroup generated by isomorphism classes of varieties over X0 with agood action2 of µ equivariant with respect tot the trivial action on X0 2 A good action of µ on V is one

that factors through some µNfor finite N and all µN-orbits are

contained in an affine open set.

moduli the relations:

[V → X0] = [V r Z → X0] + [Z → X0]

for any Z ⊂ V, a µ-invariant closed subvariety of Z, and

[X0 ×W → X0] = [X0 ×Cn → X0]

where W is an n-dimension µ representation. Cn is the trivial n-dimensional representation. In other words

[X0 ×W → X0, µ] = Ln, where L = [A1 × X0 → X0].

Mar 19

Now to do all of this in algebro-geometric setting, we use the ideaof formal arcs. We will use arcs to probe the local geometry of X andf new x0 ∈ X.Definition 5.15. We define the order n arc-space at x0 ∈ X, to be

Ln(X) = Mor(Spec(C[t]/tn+1), X)


as an affine X-scheme, Ln(X)→ X. Let f : X → C be a morphism andx0 ∈ X0 live in the central fiber. We define the order n arcs in X whichlie in X0 to order n− 1 and leaves the central fiber at order n to be thespace

Xn,1 = {ϕ ∈ Ln(X) : f ◦ ϕ = tn}.Note that Xn,1 → X0 is also an X0-scheme and it has a natural action

of µn given by ϕ(t) 7→ ϕ(e2πi/nt). So [Xn,1] ∈ MotX0 .Theorem 5.16 (Denef, Loeser). The zeta function of arcs

Z(T) =∞

∑n=1

[Xn,1]L−n dim XTn ∈ MotX0 [[T]]

is a rational function in T provided that X is smooth.Definition 5.17. The motivic Milnor fiber is finally defined as

MFf = − limT→∞

Z(T) ∈ MotX0 .

Example 5.18. Let x0 ∈ X0 be a smooth point of X0. Let (y1, · · · , yd) bevariables of coordinate ring locally at x0 such that f (y1, · · · , yd) = y1.Then Xn,1 is given by equations

ai0 + ai

1t + · · ·+ aintn, i = 1, · · · , d.

So Xn,1 → X0 is just the affine space [X0 ×C(d−1)n → X0]. Then

Z(T) =∞

∑n=1

[X0 → X0]L(d−1)n−ndTn

= [X0 → X0]L−1T

1−L−1TAs T → ∞, we conclude that over the smooth part of X0 we have

MFf = [X0 → X0]

i.e. the nearby fiber is isomorphic to central fiber over smooth points.Definition 5.19. The motivic vanishing cycle is defined as

ϕ f = [X0]−MFf ∈ Motcrit( f ) ⊂ MotX0 .

Example 5.20. Consider f (x) = x2 from affine line to itself. The spaceXn,1 are cut out by

Xn,1 = {(z0 + z1t + · · ·+ zntn)2 = t2 mod tn+1}.

This determines the arc-spaces to be,

L1 = ∅

L2 = {z0 = 0, z21 = 1, z2 ∈ C} = [two points, µ2]×L

L3 = ∅

L4 = {z0 = z1 = 0, z22 = 1, z3, z4 ∈ C} = [two points, µ2]×L

...


So we have

Ln =

{∅ n is oddµ2 ×Ln/2 n is even

Z(T) =∞

∑a=1

µ2 ×LaL−2aT2a

= µ2

∞

∑a=1

(L−1T2)a

= µ2L−1T2

1−L−1T2

Hence the motivic Milnor fiber is a copy of µ2, which should not sur-prise us because the monodromy action of x 7→ x2 is by flipping twopoints. And the motivic vanishing cycle is ϕ f = 1− µ2.

Already we see that computing the motivic Milnor fiber can be cum-bersome in most situations. Now we are going to discuss some toolsthat make this means easier.

Mar 21

5.5.1 Computing the Milnor fibers using embedded resolutions

Figure 5.1: Resolution of singularities

Let Y h−→ X be a smooth map and h is a proper resolution of singular-ities (hence being rational, dominant, and proper). We require h to bean isomorphism away from crit( f ) and h−1(X0) be a normal crossingdivisor with except finitely many components E1, · · · , En.

Recall that the fact that h is a nomral crossing divisor means that allEi are smooth and for any set of indices I ⊂ {1, · · · , n}, EI := ∩i∈I Ei

is smooth of dimension d− |I| where d = dim X.Using the notation E◦I = EI r EI ∩

(∪j 6∈I Ej

)we have

h−1(X0) = t∅ 6=I⊂{1,··· ,n}E◦I .

Let ∑ niEi be the pullback of the divisor {0} ⊂ C via f ◦ h. We let

nI = gcdi∈I

(ni).

Finally there exists an unramified cover of order nI , E◦I → E◦I so[E◦I , µnI ] ∈ MotX0 .Theorem 5.21. We can now compute the motivic Milnor fiber

MFf = ∑∅ 6=I⊂{1,··· ,n}

(1−L)|I|−1[E◦I , µnI ].

The covers E◦I → E◦I are restrictions of (ramified covers) EI → EI

and EI |EJ⊃I is a disjoint union of a bunch of copies of EJ → EJ .


Example 5.22. Back to the case of f (x) = x2 from C to C, X = Y inthis case and E0 is just a point and n0 = 2. The covering E0 → E0 is{•, •} → {•}. So we have [E0] = µ@ and MFf = µ2.Example 5.23. Given X = C3 → C via (x, y, z) 7→ xyz, the central fiberconsists of the xy, yz and xz-planes (denoted as E0

i for i = 1, 2, 3) andthe critical locus of f consists of the coordinate axis. So we have

MFf = E◦1 +E◦2 +E◦3 +(1−L)(E◦12 +E◦13 +E◦23)+ (1−L)2E123 ∈ MotX0 .

So we see that Φ f = X0 − MFf is clearly supported on the criticallocus. Let us compute the image of MFf under the map MotX0 →Motpt (that forget maps to X0).

MFf = 3([C× ×C×]) + (1−L)(3[C×]) + (L− 1)2

= (L− 1)2(3− 3 + 1) = [C× ×C×].

Example 5.24. Let f : Cn → C be given by f (x1, · · · , xn) = ∑ x2i . The

singular fiber is a cone (so not normal crossings unless n = 2).So we need to blow this up. And it turns out that the blow up at

the origin suffices. We will use the notation

Qn−1 = {x20 + · · ·+ x2

n = 0} ⊆ Pn

equipped with the Z/2-action x0 ↔ −x0.Let Y = Bl0(Cn) and E0 = X0 be the proper transform of X0. Let

E1∼= Pn−1 be the exceptional divisor of the blow up . The intersection

of these subspaces is E01 = Qn−2 ⊂ Pn−1.

Figure 5.2: Resolution of the singularity∑n

i=1 x2i = 0

E◦1 → E◦1 is the restriction of the 2:1 map E1 → E1∼= Pn−1 branched

on Qn−2 ⊂ Pn−1 (so in the notation of previous theorem, n0 = 1,n1 = 2 and n01 = 1). So [E◦1 , µ2] = [Qn−1, µ2] − Qn−2. Now wecompute the vanishing cycle

Φ f = X0 − (E◦0 + [E◦1 , µ2] + (1−L)Qn−2)

= E00 + 1− E0

0 − ([Qn−1, µ2]−Qn−2)− (1−L)Qn−2

= 1 + LQn−2 − [Qn−1, µ2] ∈ Kµ0 (Var/C).

Let p ∈ Qn be a µ2 invariant point. Then projection from p to Pn

gives us a rational map Qn 99K Pn. So we have

BlpQn

��

∼= // BlQn−2 Pn

��

(Qn, µ2) (Pn, µ2)

The right hand map contracts the quadric of lines in the tangent planeTpQn. On Qn−2 ⊂ P(T, Q), µ2 acts trivially.


So we have

[Qn, µ2]− p + Pn−1 = [Pn, µ2]−Qn−2 + P1.Qn−2

which gives us a recursive formula,

[Qn, µ2] = Pn −Pn−1 + 1 + LQn−2 = Ln + 1 + LQn−2

starting in Q0 = µ2 and Q1 = P1. We solve the recursion and get

[Qn, µ2] =

{Pn n is oddPn + L

n/2(µ2 − 1) n is even.

We conclude that the motivic vanishing cycle is

Φ f = [X0]−MFf =

{Ln/2 n is evenLn−1/2(1− µ2) n is odd

.

Mar 24

5.6 Motivic Thom-Sebastiani theorem

Let f : X → C and g : Y → C be two algebraic morphisms to C, anddefine f + g : X × Y → C via (x, y) 7→ f (x) + g(y). We will definea convolution product on MotX×Y such that the following theoremholds,Theorem 5.25.

Φ f+g = (π∗XΦ f ) ∗ (π∗YΦg) ∈ MorX×Y .

We define the convolution product on Motpt = Kµ0 (Var/C)/ extra relations

as follows.First we note that to each variety X with a µd action, we may asso-

ciate an X-bundle over C× with monodromy µd:

X×µd C× → C×

where the notation X ×µd C× is the quotient X × Cµ/µd with the di-agonal action. Conversely given such an object, we can recover (X, µd)

by taking the fiber over 1.Now, for X and Y with µ-action, we can find d such that µd acts on

both spaces. We define J(X, Y) ∈ Mot as the fiber over 1 of (X ×µd

C×)× (Y×µd C×) equipped with the diagonal µd-action.

(X×µd C×)× (Y×µd C×)

((��

C× ×C×+

// C


Definition 5.26. Give X, Y, and J(X, Y) as above, we define the convo-lution product of X and Y by

X ∗Y = −(

J(X, Y)− (L− 1)X×Yµd

).

With this definition we get X ∗ 1 = X.Example 5.27. Let us abbreviate [••, µ2] as µ2 and compute µ2 ∗ µ2.Note that µ2 ×µ2 C× → C× is isomorphic to the degree two coverC× → C× via u 7→ u2. So

J(µ2, µ2) = {(u, v) ∈ C× ×C× : u2 + v2 = 1}

where the diagonal µ2 action is (u, v) 7→ (−u,−v). But the subspaceof C×C cut out by u2 + v2 = 1, is isomorphic to [C×, µ2] which tellsus

J(µ2, µ2) = [C×, µ2]− µ2 − µ2 = [C, µ2]− 1− 2µ2 = L− 1− 2µ2.

Therefore,

µ2 ∗ µ2 = −(

L− 1− 2µ2 − (L− 1)(•, •)× (•, •)µ2

)= L− 1 + 2µ2

Corollary 5.28. This suggests that we may impose the following relation inthe ring of motives:

(1− µ2) = L12 ∈ Mot

because (1− µ2) ∗ (1− µ2) = 1− 2µ2 + L− 1 + 2µ2 = L.With this introduction we can now state the motivic version of

Thom-Sebastiani theorem:Theorem 5.29 (Thom-Sebastiani). We have the following idenities for van-ishing cycles

Φx21+···+x2

n= (Φx2)∗k

and therefore

Φx21+···+x2

n= (1− µ2)

k =

{L

k2 k is even

Lk−1

2 (1− µ2) k is odd

or if we identify (1− µ2) and L12 we get

Φx21+···+x2

n= L

k2

always.


5.7 Homogenous affine morphism–an application of embedded res-olutions

Let X ∼= Cn fd−→ C where fd is any homogeneous morphism of degreed. This means that f−1

d (0) is a cone and that f−1d (1) =: X1 has a natural

µd-action via

(x1, · · · , xn) 7→ (ξdx1, · · · , ξdxn)

where ξd∼= e

2πid .

Figure 5.3: Homogenous map of degreed

We define the following integration map∫

X0: MotX0 → Mot as the

push-forward map π∗ where π : X0 → pt. Then we haveProposition 5.30. ∫

X0

MFfd= [ f−1

d (1), µd].

Proof. As first step we blow up Cn in the origin, and let X0 be theproper transform of central fiber X0. Let locus { fd = 0} now intersectsthe exceptional projective (n− 1)-plane, which sits as the zero sectionof the line bundle OPn−1(−1) ∼= Bl0(Cn). We now form the embeddedresolution of this singularity as shown in figure 5.4. Note

Figure 5.4: Blow up of the cone (themiddle fibration) constructs X0 overthe central fiber X0; the exceptional

divisor of X0 is the base variety ofthis fibration, { fd = 0} ⊆ Pn−1 in

the lower middle picture. This is fol-low by the embedded resolution ofsingularities, E0, consisting of nor-mal crossing divisors F1, · · · , Fn inlower right hand picture; The pull

back (upper right hand picture) createsE1, · · · , Ek and E0 ⊆ Pn−1 on the base.

that Ei → Fi∼= Ei ∩ E0 is a C-fibration. And likewise, for any set of

indices I, EI → FI∪{0} is a C-fibration. Also note that E0I → FI∪{0} is a

C×-fibration and hence

E0I = (L− 1)E0

I∪{0}

and after pulling back along f , we get

[E0I , µnI ] = (L− 1)[EI∪{0}, µnI∪{0}].


Now the integration map send the Milnor fiber to∫X0

MFf = ∑∅ 6=J⊆{0,··· ,k}

(1−L)|J|−1[E0J , µnJ ]

= ∑∅ 6=I⊂{1,··· ,k}

(1−L)[E0I ] + (1−L)|I|[E0

I∪{0}] + [E00, µd]

= [E00 , µd]

Now we study the motivic class of E00 . Note that E0

0 → E00 = Pn−1 −

{ fd = 0} is the restriction of the d-fold branched covering of Pn−1

branched along { fd = 0}.We can view fd|Pn−1 as a section of O(d) → Pn−1. So the equation

fd = sd for s ∈ C, cut out a subvariety of O(1)→ Pn−1:

ι : E00 = { fd = sd, s 6= 0} → O(1)r {zero section}.

Here the ambient space O(1)r {zero section} has an action of µd vias 7→ ξds. We may compose the injection ι with the morphism

O(1)r {zero section} → O(−1)r {zero section}

via (x1, · · · , xn, s) 7→ (x1, · · · , xn, u = 1s ). This idenitifies

{ fd = sd, s 6= 0} = { fd = u−d, u 6= 0} ∼= {ud fd(x1, · · · , xn) = 1}

but note that fd is written in homogeneous coordinates, and thereforefd(x1, · · · , xn) = fd(ux1, · · · , uxn) hence we conclude that∫

X0

MFfd= [E0

0 , µd] = [ f−1d (1), µd].

Mar 26

5.8 Generalized DT-invariants of Kontsevich and Soibleman

Let M be the moduli stack of objects in Coh(X) where X is a Calabi-Yau threefold. Recall that given an object E on Coh(X), M is locallywritten as {d fE = 0}/ Aut(E) where fE : Ext1(E, E) = Def(E) → C isthe superpotential. Notice that Lie(Aut(E)) ∼= Ext0(E, E).Theorem 5.31 (Kontsevich-Soiblemann). There exists a class

[W →M] ∈ MotM = Kµ0 (St/M) = Kµ

0 (Var/M)[L−1, (1−Ln)−1]n=1,2,···

such that on each local model {d fE = 0}/ Aut(E), W is given by

L−12 (dim Ext1(E,E)−dim Ext0(E,E))Φ fE . (5.5)


Notice how this suggests consis-tancy with Behrend’s weighted

euler characteristic, ν(E):

(−1)ext1(E,E)−ext0(E,E)(1− e(MFf (0))).

As a matter of fact, W → M

is called motivic weight function.

Remark 5.32. Recall that f : V → C with Φ f = [V]−MFf ∈ Motcrit( f )and that

L12 = (1− µ2) ∈ Motpt

e(L12 ) = e(1− 2pt) = −1

Therefore the euler characteristic of W →M over a point E is

e(W|E) = (−1)dim Ext1(E,E)−dim Ext2(E,E)(1− e(MFfE(0))).

Ignoring stackiness and automorphisms, the content here is the fol-lowing: at E we have the local model given by crit( fE), and for aneighbor point E′ ∈ M we have a different local model described viacrit( fE′) = {d fE′ = 0} ⊂ Ext1(E′, E′). However the tanget space of Mis given by Ext1(E, E) so we may view E′ ∈ Ext1(E, E). If we expand fE

about E′, we then get a function beginning with quadratic terms. Thismeans that E′ ∈ crit( fE). So after a non-linear change of coordinateson Ext1(E, E) we get

fE|centered at E′ =

in dim Ext1(E, E) variables︷︸︸︷fE′ + homogeneous quadratics terms︸︷︷︸

dim Ext1(E,E)−dim Ext1(E′ ,E′) variables

.

So we apply the motivic Thom-Sebastiani here

Φ fE |centered atE′ = Φ fE′∗ (L

12 )dim Ext1(E,E)−dim Ext1(E′ ,E′).

The factor of L1/2 in front of 5.5 compensates for this difference and

W →M is independent of choices of local models.Although there is one important subtly here. We used Φx2

1+···+x2n=

(L12 )n, which we proved in Motpt. However, In the above argument

we need this to be true for families of quadratics over M. To get anappropriate L

12 ∈ MotM we need a square root

√D of

D = sdet(Ext•(E, E))→M.

Morally speaking,

D|E =top∧

Ext0(E, E)⊗top∧

Ext1(E, E)∨︸︷︷︸⊗top∧

Ext2(E, E)⊗top∧

Ext3(E, E)∨︸︷︷︸where the two parts shown are isomorphic. So if Exti(E, E) were lo-cally free then the square root

√D existed canonically.

Theorem 5.33 (Kontsevich-Soiblemann). Let H(C) be the Hall algebra ofa Calabi-Yau category C, endowed with orientation data (and consequently[W →M]). Then we have a homomorphism of algebras over MotM

Φ : H(C)→ MotM [[qK(C)]]


where elements of MotM [[qK(C)]] are ∑α∈K(C) mαqα for mα ∈ MotM with

qα.qβ = L12 〈α,β〉qα+β.

Here 〈α, β〉 = ∑3i=0(−1)i dim Exti(α, β) is a skew-symmetric pairing.

Specifically,

Φ([X π−→M]) = ∑α∈K(C)

[Xα ×M W]qα

where Xα = π−1(Mα). In this manner Φ gives the generating function ofwhat we now define as motivic DT-invariants.

Mar 28

Last time we discussed1. The existence of motivic weights W → M ∈ Mot)M, where M is

the moduli stack of objects

W|E = L−12 (Ext1(E,E)−Ext0(E,E))Φ fE

where fE : Ext1(E, E) → C is a local model for the moduli space.Here also

L12 = (1− µ2).

2. There is a homomorphism

Φ : H(C)→ Mot [[qK(C)]].

The image ring is the collection of all elements ∑α∈K(C) mαqα withthe operation

qα.qβ = L12 〈α,β〉qα+β

via 〈α, β〉 = χ(α, β). The integration map is defined such that

Φ([X →M]) = ∑α∈K(C)

Xα ×M Wqα

where Xα = π−1(M(α)).Example 5.34. Let I = [I•(X, •)→M] via [OX � OC] 7→ OC where

I•(X, •) =⊔n,β

In(X, β).

We haveΦ(I) = ∑

n,β[In(X, β)×M W]qnvβ.

This defines for us the motivic Donaldson-Thomas invariants3 3 Strictly speaking we want the weightsassociated to In(X, β), not M.

ZDT,motX = Φ(I).


If we apply the Poincare polynomial to the image of integration, weget the refined invariants

ZDT,refX = Pt(Φ(I)).

To get the numerical invariants we further apply the euler characteris-tic:

ZDT,numX = e(Pt(Φ(I)))

when this is possible! This is a composition through the followingrings

Mot [[qK(C)]]Pt−→ Z[t, t−1, (1− tm)−1][[qK(C)]] e−→ Z[[qK(C)]].

5.9 Applications

5.9.1 Finishing Bridgeland’s proof

Recall that we derived identity 5.4 in the Hall algebra of C = Coh≤1(X)

for a Calabi-Yau threefold X:

I ∗ 1P ∗ 1Q = I0 ∗ 1P ∗ PT ∗ 1Q. (5.6)

Now we apply our integration map. We get

ZDT,mot0 .Φ(1P).ZPT,mot.Φ(1Q) = ZDT,motΦ(1P)Φ(1Q) ∈ Mot [[qK(C)]].

But because 〈•, •〉 is zero on K(C) the quantum torus is commutativein this case (note that through Serre duality, 〈•, •〉 works like Poincarepairing). This completes the proof of

ZPT,mot =ZDT,mot

ZDT,mot0

.

This in particular holds for refined and numerical invariants.

5.9.2 Degree zero motivic DT invariants

Let C = Coh≤0(C3). Understanding this allows us to understand

Coh≤0(X) for any Calabi-Yau threefold. Firstly Note that Mn is atthe same time moduli stack of the following categories• zero dimensional coherent sheave fo lengths n on C3;• modules W over C[x, y, z] with dimC W = n;• triples (x, y, z) of endomorphisms of Cn as vector space such that

[x, y] = [x, z] = [y, z] = 0;• representations of the following quiver with

• xvv

y

��

z66

relations [x, y] = [x, z] = [y, z] = 0;


• flat rank n vector bundles on T3.We will work with the most concrete one:

Mn = {(x, y, z) ∈ (Mn×n(C))3 : [x, y] = [x, z] = [y, z] = 0}/ GLn .

Let fn : M3n×n → C be given via (x, y, z) 7→ tr(x[y, z]). Then

{d fn = 0} = {[x, y] = [x, z] = [y, z] = 0}

and Mn = {d fn = 0}/ GLn and in fact this is the local model of Mn

at the origin, i.e. at E = O⊕n0 . Here O0 is the skyscraper sheave at the

origin and one can check that

fn : Ext1(O⊕n0 , O⊕n

0 ) ∼= M3n×n → C

is the same as the above expression. In fact this is a global descriptionof Mn in this case as well.

Our motivic invariant is

ZM = Φ(1M) = ∑n[Wn]qn

=∞

∑n=0

[L−12 (3n2−n2)Φ fn ]/[GLn]qn

because dim Ext1 = 3n3 and dim Ext0 = n2. We now need to find thevanishing cycle of fn. Note that by the expression of fn : (Cn2

)3 → C,it is a homogeneous polynomial of degree 3. Hence,

Φ fn = [ f−1n (0)]− [ f−1

n (1), µ3]

= L−n2([ f−1

n (0)]− [ f−1n (1)])

Thus

f−1n (0)− f−1

n (1) = f−1n (0)− 1

L− 1f−1n (C×)

=1

L− 1((L− 1) f−1

n (0)− (L3n2 − f−1n (0)))

=1

L− 1(L f−1

n (0)−L3n2)

Now we find the motivic class of f−1n (0) by stratifying the zero locus

of tr(x[y, z]) in two strata: one, if [y, z] = 0, in which case x can beanything, and the other one, where [y, z] 6= 0 and x is orthogonal to it.So we have

f−1n (0) = Ln2

Cn + Ln2−1 (L2n2 − Cn)︸︷︷︸[y,z] 6=0

= Ln2−1(LCn + L2n2 − Cn)


where Cn is the motive of the commuting endomorphisms. We have

L−n2( f−1

n (0)− f−1n (1)) = Cn.

So we get

ZM(q) =∞

∑n=0

[Cn]

[GLn]qn.

This series was computed by Feit-Fine in 1940! They were interested inthe question of finding the probability that two n× n matrices over Fp

commute? Their proof amounts to stratifying the variety Cn by strataisomorphic to A

kiFp

. For them #[AFp ] = p, hence if we substitute L forp in their formula we get the beautiful formula

∞

∑n=0

[Cn]

[GLn]qn =

∞

∏m=1

∞

∏j=0

(1−L1−jtm)−1∈ Mot [[t]] ⊂ K0(Var)[[L−1]].

Mar 31

Last time we considered C = Coh≤0(C3). The motivic weight

Wn → Mn here is L−n2Φ fn = Wn. We computed, ZM, the motivic

DT-invariants of M and quoted a result of Feit and Fine

ZM =∞

∏m=1

∞

∏j=0

(1−L1−jtm)−1∈ Z[L][[L−1, q]].

In following sessions we give a proof of this using power structures.Up to second order terms we have

ZM =∞

∏j=0

(1 + L1−jq + o(q2))

= 1 + (∞

∑j=0

L1−j)q + o(q2)

So the coefficient of q is ∑∞j=0 L1−j = L2

L−1 which is, as expected,[C1]/[Gl1]. Note that M is a stack (not a scheme) and therefore thespecialization L→ 1 is not well-defined.

Traditional DT invariants of C3 come from Hilbn(C3) = I0(C3, 0):

NDTn = evir(In(C

3, 0)).

As per previous notation, let

I0 = [tn In(C3, 0)→Mn] ∈ H(C)

where the structure morphism over Mn, maps OC3 � OZ to OZ ∈ Mn.Also let

IOM = [M(O)→M]


via (O → F ) 7→ F . Recall that the content of Hall algebra identityIOM = I0 ∗ IM is that every morphism factors as a surjection followed

by an injection:

O

��~~

0 // OZ // F // Q // 0.

Note that Mn(O) → Mn is a fibration with fiber Cn and henceMn(O)×Mn W is a vector bundle over Wn, therefore:

[Mn(O)×Mn W] = Ln[Wn].

Consequently Φ(1OM) = Zm(Lq) and

Zm(Lq) = Φ(I0)Zm(q) ∈ Mot [[q]]

where Mot[[q]] is commutative. So we have

Φ(I0) =Zm(Lq)Zm(q)

.

The formula

Φ(I0) =∞

∑n=0

[In(C3, 0)×Mn Wn]qn

is close to

ZDT,mot0 (C3) =

∞

∑n=0

[Wn → In(C3, 0)]qn

where Wn is the weight function coming from In.So we look at Ext•(O → OZ, O → OZ) versus Ext•(OZ, OZ). These

differ by Ext0(OZ, OZ) = Cn only by automorphisms. Hence we have

[Wn → In(C3, 0)] = L−

n2 [In(C

3, 0)×Mn Wn]

and therefore

ZDT,mot0 (C3) = Φ(L−

12 q) =

Zm(L12 q)

Zm(L12 )

.

So we apply

Zm(q) =∞

∏m=1

∞

∏j=0

(1−L1−jqm)−1

to it and we get

ZDT,mot0 (C3) =

∞

∏m=1

∞

∏j=0

(1−L1−j+m/2qm)−1

(1−L1−j−m/2qm)−1

=∞

∏m=1

m−1

∏j=0

(1−L1−j+m/2qm)−1


and remember that this can be used to find specializations to refinedand numerical invariants:

ZDT,ref0 (C3) =

∞

∏m=1

m−1

∏j=0

(1− t2−2j+mqm)−1

ZDT0 (C3) =

∞

∏m=1

m−1

∏j=0

(1− (−q)m)−1

=∞

∏m=1

(1− (−q)m)−m = M(−q).

and notice we have not used the torus action at all. So this can beregarded as an independent proof for computation of this numericalinvariant without using the box counting argument.

We can use this method to compute invariants for objects in thewhole category, but before doing that we present another examplefirst.Example 5.35. Let C be a curve of genus g in a Calabi-Yau threefoldX. For example we can take a curve C and consider the total space ofa generic rank two vector bundle N → X such that ∧2N ∼= KC. ThePT-moduli space is

PTn(X, [C]) =

{non-zero OX → s∗L,

L→ C is line bundleand n = deg L + 1− g

}= Symn+g−1(C).

hence smooth of dimension n + g − 1. The moduli space is unob-structed, and the local Chern-Simon potential

f : Ext1(O→s∗L, Ox → s∗L)→ C

is zero. The motivic weight is L−1/2(n+g−1). So

ZPT,mot[C] =

∞

∑n=1−g

L−1/2(n+g−1)[Symn+g−1(C)]qn

ZPT,ref[C] (X) =

∞

∑n=1−g

t−(n+g−1)Pt(Symn+g−1(C))qn+g−1q1−g

q1−g∞

∑k=0

Pt(Symk(C))(t−1q)k

Here the Poincare polynomial is the power series of Betti numbersPt(s) = ∑` b`s`. Now we use this theorem of MacDonald:Theorem 5.36 (MacDonald).

∞

∑k=0

Pt(Symk S)zk =∏odd `(1 + t`z)b`

∏even `(1− t`z)b`


So∞

∑k=0

Pt(Symk(C))zk =(1 + tz)2g

(1− z)(1− t2z).

and we get the refined invariants

ZPT,ref[C] (X) =

(1− q)2gq−g

q−1(1− tq)(1− t−1q)

=(q−1+ q + 2)g

(q + q−1)− (t + t−1)

and notice that this is symmetric under q ↔ q−1 and we can do theMNOP substitution in it. Also by letting t→ −1 we get the numericalPT-invariants to be (q1/2 + q−1/2)2g−2.

April 1

5.10 Power structures and their applications

Last time we computed ZDT,mot0 (C3), i.e. virtual motives of Hilbn(C3).

We now want to use a power structure on K0(Var/C) to compute ZDT,mot0 (X)

where X is any Calabi-Yau threefold.Definition 5.37. A power structure on a commutative unital ring R isa map

R× (1tR[[t]])→ 1 + tR[[t]]

given by (m, A(t)) 7→ A(t)m, satisfying1. A(t)0 = 1,2. A(t)1 = A(t),3. (A(t)B(t))m = A(t)mB(t)m,4. (A(t))m+n = A(t)m A(t)n,5. (A(t))mn = ((A(t))m)n,6. (1 + t)m = 1 + mt + o(t2),7. A(tk)m = (A(s)m)|s=tk where k is a natural number.

More or less this is equivalent to a pre-lambda ring structure on R.Theorem 5.38 (Gusein-Zade, Melle-Hernandez). There exists a powerstructure on K0(Var/C) uniquely determined by

(1− t)−[X] =∞

∑n=0

[Symn X]tn.

So note that number (7) in the definition implies that

(1− t2)−[X] = ∑[Symn X]t2n

but it is not true that

(1 + t)−[X] =∞

∑n=0

(−1)n[Symn(X)]tn.


More generally, if A(t) = ∑ Aiti where Ai’s are varieties and X is avariety, then

(A(t))X =∞

∑k=0

Bktk

where Bk is the space of configurations of Ai-valued points on X oftotal charge k, i.e. set of pairs (S, ϕ) where S ⊂ X is a finite subset andϕ : S→ ti Ai, such that k = ∑x∈S i(ϕ(x)). We can write this as

Bk =⊔α`k

(∞

∏i=1

Xbi(α) r ∆

)×Sα

(∞

∏i=1

Abi(α)i

)

where bi(α) is the number of i’s in the partition α and Sα = ∏∞i=1 Sbi(α)

.Lemma 5.39 (Totaro). In K0(Var/C) we have [Symn Ck] = [Cnk]. So interms of power structures this means that

(1− tm)−Lk= (1−Lktm)−1.

Aside: Under the Poincare polynomial specialization K0(Var/C)→Z[s] we have L 7→ s2. Henc the power structure specializes to a powerstsructure on Z[s] satisfying

(1− t)−(−s)k= (1− (−s)kt)−1.

This helps us to recover the formula of MacDonald as follows:

∞

∑n=0

[Symn X]tn = (1− t)−[X]

∞

∑n=0

Ps(Symn X)tn = (1− t)−∑ bksk

= ∏k(1− t)−bk(−1)k(−s)k

= ∏k(1− (−s)kt)−bk(−1)k

=∏k is odd(1 + skt)bk

∏k is even(1− skt)bk

= (1− t)−e(x) if we take euler characteristics by s = −1.

April 4

Last time we introduced power structures:

R× (1 + tR[[t]])→ 1 + tR[[t]].

So when R = K0(Var/C), there exists a power structure such that ifA(t) = ∑∞

i=0 Aiti, with Ai being varieties, then for a variety X, we have

A(t)X = ∑ Biti


where informally speaking Bk is the space of systems of particles oftotal charge k where the internal state space of a charge i particular isAi, and ∑ i` = k. For example we saw that

(1 + t + t2 + ·)[X] = (1− t)−[X] = ∑k

Symk Xtk

and we found(1− t)−Lk

= (1−Lkt)−1.

What we want is not just a power structure on the Grothendieckgroup of varieties, but more generally that of stacks where motives ofgeneral linear groups are inverted.Lemma 5.40. The power structure on K0(Var/C) extends to K0(Var/C)[[L−1]],such that (1− t)−L−k

= (1−L−kt)−1 and such that if A(t) = ∑∞i=0 Aiti

where Ai are now stacks, and X is still a variety, then the above characteri-zation still holds; namely, A(t)X = ∑ Biti, where Bi is the configuration ofAi-valued point son X as before.

Note that K0(Var/C)[[L−1]] contains K0(St/C) = K0(Var/C)[L−1, (1−Ln)−1].

5.10.1 Application 1 - Feit-Fine formula

As an application we will now give a simple proof of Feit-Fine formula:recall that in our notation Cn is the scheme of commuting matrices{[A, B] = 0} ⊂ C(2n2), and Cn/ GLn ∼= Mn(C2) is the stack of sheavesof dimension zero and length n on C2. We define

C(t) =∞

∑n=0

[Cn]

[GLn]tn = ∑

n=0∞[

Cn

GLn

]tn.

This means that

C(t) =∞

∑n=0

[Mn(C2)]tn

=

(∞

∑n=0

[M(0,0)n (C2)]tn

)[C2]

Here the notation M(x0,y0)n (C2) is for the space of sheaves supported at

the point (x0, y0). Note that this means multiplication by (x− x0)N and

(y− y0)N is zero for large enough N. Hence A− x0 id and B− y0 id

are nilpotent, and eigenvalues of A are all x0 and eigenvalues of B areall y0.

So we have(∞

∑n=0

[M(0,∗6=0)n (C2)]tn

)=

(∞

∑n=0

[M(0,0)n (C2)]tn

)[Cr{0}]


and hence

C(t) =

(∞

∑n=0

[M(0,∗6=0)n (C2)tn

) L2L−1

.

Note that

M(0,∗6=0)n (C2) = {[A, B] = 0, A is nilpotent and B is invertible }/ GLn

=⊔

λ`n

{Jλ = BJλB−1}/ StabJλ(GLn)

and here [M(0,∗6=0)n (C2)] is a finite number of copies of motives of

points, and the cardinality of it is p(n), the number of partitions ofn. So now we finish with

C(t) =

(∞

∑n=0

p(n)tn

) L2L−1

=

(∞

∏m=1

(1− tm)−1

)∑∞k=1 L2−k

=∞

∏k=1

∞

∏m=1

(1− tm)−L2−k=

∞

∏k=1

∞

∏m=1

(1−L2−ktm)−1.

5.10.2 Application 2 - motivic invariants of Hilbert schemes of points

As we promised, we can now compute the motives of Hilbert schemesof points on smooth X of dimension d. We start with

∞

∑n=0

[Hilbn(X)]tn =

(∞

∑n=0

[Hilbn(0)(C

d)]tn

)[X]

=

(∞

∑n=0

[Hilbn(Cd)]tn

)[X].L−d

For a Calabi-Yau threefold X we now have,

ZDT,mot0 (X) =

∞

∑n=0

[W → In(??)]tn

but W is also local in nature so

ZDT,mot0 (X) =

(∞

∑n=0

[W → In(C3, 0)]tn

)[X]L−3

=

(∞

∏m=1

m−1

∏j=0

(1−Lm/2+1−k)

)[X]L−3

So we get that

ZDT,mot0 (X) =

∞

∏m=1

m−1

∏j=0

(1−Lm/2+1−k)[X].


Now for specializations we have the Poincare polynomial

Ps[X] =6

∑d=0

bdsd =6

∑d=0

bd(−1)d(−s)d

so as we substitute L1/2 with s we get the refined invariants

ZDT,ref0 (X) =

6

∏d=0

∞

∏m=1

m−1

∏j=1

(1− (−s)m−4−2j(−t)m

)(−1)dbd(−s)d

=6

∏d=0

Md−3(−s,−t)(−1)dbd

where

Mδ(s, t) =∞

∏m=1

m

∏j=1

(1− sm−2j+1+δtm

)−1.

This is the analogue of MacDonald and Göttche’s formula for Poincarepolynomial of Hilbert schemes of n points over curves and surfaces.It is not clear if this picture can go on in dimension four, as thoseschemes do not have a virtual fundamental class the way we need.

April 7

5.10.3 Case of local curves

Now we want to consider local curves X = tot(L1 ⊕ L2) where L1 ⊕L2 → C is a rank two bundle over a smooth curve of genus g and L1⊗L2 ∼= KC. Solving the curve-counting invariants of local curves is animportant but hard problem (but doable)! We will find ZPT/DT/GW

d[C] (X).

We will specifically compute ZDTd[C](tor(K1/2⊕K1/2) for these local curves.

Park and Ionel, show that on a generic almost complex structure ona Calabi-Yau threefold, the pseudo-holomorphic curves are “almost”isolated. And using this they show that the general Gupakumar-Vafaconjecture reduces to local curves with L1 = L2 = K1/2

C . One can thinkof such local geometries, as the idealized neighborhood of a curve in ageneral Calabi-Yau threefold.

Using power structures we obtain an easy proof4 of the formula for 4 By ignoring some Behrend functionissues.

ZDTd (X) = ∑

ne(In(X, d][C]), νn)qn

where νn : In(X, d[C]) → Z is the Behrend function and recall thatBehrend shows that νn only depends on In(X, d[C]) as a scheme. Xhas a T = C× × C×-action. So T acts on In(X, C) and given that νn

only depends on the scheme structures, it has to be constant on orbitsof T. So we may stratify In(X, d[C]) by dimension of orbits, and all the


ones that are not fixed-points have zero weighted euler characteristics,so we have

ZDTd (X) = ∑

ne(In(X, d[C])T , νn)qn.

What do T-invariant subschemes of X look like? It is a d-fold thick-ening of the curve supported on the zero section and it has embeddedpoints and its generic slice with a fiber is a T-invariant of C2, hencecorresponding to a monomial ideal, hence corresponding to a partitionα ` d of integer d. We use the notation (i, j) ∈ α for (i, j) ∈ Z⊕2

≥0 if andonly if xiyj ∈ OZ = C[x, y]/IZ where IZ is the associated monomialideal.

Pure subschemes which are T-invariant are in bijection with α ` d.Note that we may write the scheme structure of X as

X = SpecC(Sym•(L∨1 ⊕ L∨2 )).

So for a pure, T-invariant subschemes Cα corresponding to some par-tition α ` d, the structure sheaf is

OCα=

⊕(i,j)∈α

L−i1 ⊗ L−j

2 .

A general T-invariant subscheme Z will have an ideal sheaf, IZ, fittingin a short exact sequence

0→ IZ → ICα→ P→ 0

where P is supported at points, and n = χ(OCα) + (length)(P). So thus

get

ZDTd (X) = ∑

ne(In(X, d[C])T , νn)qn

= ∑α`d

qχ(OCα )∞

∑k=0

e(Ik(X, Cα)T , νn)qk

here Cα is the underlying pure subschemes, and k is the length of theembedded points. The claim is that νn = (−1)n on Ik(X, Cα)T .

If we let Q = −q then we get

ZDTd (X) = ∑

α`dQχ(OCα )

∞

∑k=0

e(Ik(X, Cα)T)Qk.

The motive of Ik(X, Cα)T satisfies a power structure relation:

∞

∑k=0

[Ik(X, Cα)T ]tk =

(∞

∑k=0

[Ik(C3, Cα)

T0 ]t

k

)[C]

.

Here the subscript 0, encodes the requirement for embedded points tobe only supported at the origin. We derive that

ZDTd (X) = ∑

α`dQχ(OCα )

(∞

∑k=0

e(Ik(C3, Cα)

T0 )Q

k

)2−2g

.


We compute the inside sum now

∞

∑k=0

e(Ik(C3, Cα)

T0 )Q

k =∞

∑k=0

e(Ik(C3, Cα)

C××C××C×)Qk

=∞

∑k=0

#

ways of adding k boxesto configuration of boxesalong x axis of profile α

Qk

But not that this is the familiar topological vertex V∅∅α(Q), and by theformula of Vafa-Okounkov-Reshetikhin it is

V∅,∅,α(Q) = M(Q) ∏(i,j)∈α

11−Qhα(i,j)

where hα(i, j) is the hook length of partition α at node (i, j). We there-fore have

ZDTd (X) = ∑

α`dQχ(OCα )(V∅∅α(Q))2−2g

ZDT0 (X) = M(−q)e(X) = M(Q)2−2g

ZDTd (X)

ZDT0 (X)

= ZPTd = ∑

α`dQχ(OCα )

∏(i,j)∈α

11−Qhα(i,j)

2−2g

We need the Euler characteristic as well given that L1∼= L2 ∼= K1/2

C ,

χ(OCα) = χ(

⊕(i,j)∈α

L−i1 ⊗ L−j

2 )

= ∑(i,j)∈α

−i deg L1︸︷︷︸g−1

−j deg L2︸︷︷︸g−1

+1− g

= (1− g) ∑(i,j)∈α

i + j + 1 = (1− g) ∑(i,j)∈α

hα(i, j)

So we get that

ZPTd = ∑

α`dQ(1−g)∑(i,j)∈α i+j−i

∏(i,j)∈α

1(1−Qhα(i,j))2

1−g

= ∑α`d

∏(i,j)∈α

(Qh(i,j)/2 −Q−h(i,j)/2)−(2−2g)

= ∑α`d

(dimQ α

d!

)2−2g


The Gromov-Witten theory is now computed via substitution Q = eiλ:

ZGWd = ∑

α`d

(∏�∈α

2 sinh(�)λ

2

)2g−2

= ∑α`d

(∏�∈α

hα(�)λ

)2g−2

(1 + o(λ2))

= λd(2g−2) ∑α`d

(d!

dim α

)2g−2+ · · ·

The last identity comes from

∏�∈α

hα(�) =d!

dim α

where dim α is the dimension of α ` d when viewed as a representationof Sd.

So the leading term in ZGWd is

λd(2g−2) = λ2g−2

where n is the genus of the domain. This corresponds to unramifiedcovers of degree d. The number of unramified covers of degree d of Cis given via

∑α`d

(d!

dim α

)2g−2.

So for genus one, we get the familiar, p(d), number of coveringsof degree d. And in genus zero, we get 1

(d!)2 ∑α`d(dim α)2. From el-ementary representation theory, if you take any group, and take allrepresentations and sum the squares of their dimensions you get theorder of the group, so we get that this is equal to 1

d! .The punch line is the interesting fact that while studying the Gromov-

Witten invariant and deformations of curves, the same Q-deformationsof integers occur, in sense of classical deformations of integers.

Appendix ASolution to some problems

A.1 Homework set 2, Problem 3

The goal is to analyze I3(X, 2[P1]) and I4(X, 2[P1]) in case of X =

tot(O(−1)⊕O(−1)→ P1) being the resolved conifold.Remember that for [IZ] ∈ In(X, 2[P1]), Z ⊂ X must have its curve

component be supported on P1 ⊂ X. Because

Xp−→ {xy = wz} ⊂ C4

is a resolution of singularities. But no subscheme of this affine targetspace is compact. So the scheme-theoretic image of Z, p(Z) has to besupported on points, and curve part of Z being supported on p−1(0).So Z is a length 2 thickening of P1 with embedded points. So for thegeneric fiber:

Z ∩C2 = Spec(

C[x, y]/(x2, xy, y2, αx + βy))

So the primary component of Z is determined by a subline-bundleL ↪→ O(−1)⊕ O(−1). So Z ⊂ L ⊂ X and Z is the zero section of L,hence a Cartier divisor, cut-out by exactly one equation, hecne Z =

2[P2] the unique thickening of P1 determined by L. What is χ(OX)?And what are the possible subline-bundles that can happen?

Let’s say L = O(−k). Then L → O(−1) ⊕ O(−1) are given byH0(P1, O(k − 1) ⊕ O(k − 1)), but H0(P1, O(k − 1) ⊕ O(k − 1)) = C2k

for k ≥ 1.In general, let S be a smooth surface D ⊂ S. Then by Riemann-Roch

χ(OD) =∫

Sch(OD)td(TS).

By adjunction formula

0→ OS(−D)→ OS → OD → 0


and ch(OD) = 1− e−D = D− D2

2 and

χ(OD) =∫

S(D− D2

2)(1− KS

2+ · · · ) = −D

KS2− D2

2

giving the reincarnation of adjunction formula for any surface:

χ(OD) =−12

(D2 + D.KS).

(We need compactness, but one can always compactify the surface andthen everything is supported away from the compactification locus).

We apply this to our surface L = O(−k) ⊂ X. So [P1]2 = −k andD = 2[P1] and D2 = −4k. Also KS = O(k) ⊗ KP1 = O(k − 2). SoKS.P1 = k− 2 and KS.D = 2k− 4. So

χ(OD) =−12

(−4k + 2k− 4) = k + 2.

Now I3(X, 2[P1]) is the moduli of thickenings along O(−1) ↪→ O(−1)⊕O(−1)

In case of n = 4 the space I4(X, 2[P1]) has two strata of dimension3 and 4: P3 ⊃ P1 × P1 and BlP1×P1(P1 × X) that meet in the inter-section P1 × P1. The P3 = P(H0(P1, O(1) ⊕ O(1))) corresponds toinjections of bundles 0 → O(−2) → O(−1)⊕O(−1) and the blow upcorresponds to short exact sequences 0→ O(−2)→ O(−1)→ Op → 0.

Bibliography

[] Kentaro Hori, Sheldon Katz, Albrecht Klemm, RahulPandharipande, Richard Thomas, Cumrun Vafa, RaviVakil, and Eric Zaslow. Mirror symmetry, volume 1 of ClayMathematics Monographs. American Mathematical Society,Providence, RI; Clay Mathematics Institute, Cambridge,MA, 2003. With a preface by Vafa.

[] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandhari-pande. Gromov-Witten theory and Donaldson-Thomastheory, I. ArXiv Mathematics e-prints, December 2003.

[] R. Pandharipande and R. P. Thomas. Stable pairs and BPSinvariants. ArXiv e-prints, November 2007.

[] R. Pandharipande and R. P. Thomas. The 3-fold vertex viastable pairs. ArXiv e-prints, September 2007.

[] R. Pandharipande and R. P. Thomas. Curve counting viastable pairs in the derived category. Inventiones Mathemat-icae, 178:407–447, May 2009.

[] R. Pandharipande and R. P. Thomas. 13/2 ways of count-ing curves. ArXiv e-prints, November 2011.

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