Seville|July 18th, 2012arXiv:1207.4202 X. Liao:Chern classes of logarithmic vector elds...

Chern classes of hypersurface arrangements

Paolo Aluffi

Florida State University

Seville—July 18th, 2012

Paolo Aluffi (Florida State University) Chern classes of hypersurface arrangements Seville—July 18th, 2012 1 / 29

References:

—: Chern classes of hyperplane arrangementsarXiv:1103.2777, IMRN, 2012.

—: Chern classes of free hypersurface arrangementsarXiv:1201.5396, Journal of Singularities, 2012

— & E. Faber: Splayed divisors and their Chern classesarXiv:1207.4202

X. Liao: Chern classes of logarithmic vector fieldsarXiv:1201.6110, Journal of Singularities, 2012

X. Liao: Chern classes of logarithmic vector fields forlocally-homogeneous free divisorsarXiv:1205.3843.


Summary.

Aim: To better understand the Chern classes of the complement of ahypersurface in a nonsingular variety. Prototype: Normal crossingdivisors.

So far, best results are for certain free divisors, e.g., free hyperplanearrangements.

Neat, natural-looking formula:

Theorem

Let D be a free divisor in a nonsingular variety V , satisfying certainadditional hypotheses. Then c(DerV (− logD)) ∩ [V ] = cSM(V r D).

Here, cSM is the ‘Chern-Schwartz-MacPherson’ class.

This is not true for arbitrary free divisors.

It is true for e.g. locally quasi-homogeneous free divisors in Pn

(X. Liao)


DerV (− logD) is the sheaf of ‘logarithmic derivations’ along D.D: hypersurface F = 0.∂ ∈ DerV is logarithmic w.r.t. D if ∂(F ) ∈ (F ).I.e.: there is an exact sequence

0 // DerV (− logD) // DerV // OD(D) // OJD(D) // 0

where JD ⊆ D is the subscheme defined by the partials of D.JD is the ‘singularity subscheme’ of D.

Def. =⇒ DerV (− logD) is torsion free, reflexive.

Dual: DerV (− logD)∨ = Ω1V (logD), logarithmic 1-forms.

Normal crossing divisors: Deligne, ’60s.Arbitrary divisors: K. Saito, ’80.


Intuition: A vector field on V is logarithmic if it is tangent to D at smoothpoints of D.


Definition

A divisor D is free if DerV (− logD) is locally free.

Example: D nonsingular.

Example: D simple normal crossings.

More sophisticated examples: several types of discriminants ofhypersurface singularities are free. (Saito’s motivation.)

Thoroughly studied: free hyperplane arrangements (Terao).

Some names: Aleksandrov, Buchweitz, Calderon-Moreno, Castro-Jimenez,Damon, Denham, Faber, Granger, Mond, Mustata, Narvaez-Macarro,Nieto-Reyes, Saito, Schenck, Schulze, Silvotti, Tanabe, Terao, Yoshinaga,. . . (many more! including everyone at this workshop)

Criterion: D is free ⇐⇒ D is nonsingular or JD Cohen-Macaulay,codimension 2 in V .


Theme: One has ‘better access’ to the complement of a divisor D when Dis free (+possibly other conditions).

Main instance of this phenomenon:

‘Logarithmic comparison theorem’(Castro-Jimenez, Narvaez-Macarro, Mond):

D ⊆ V locally quasi-homogeneous free divisor, U = V r D, j : U → V .

Then the natural morphism Ω•V (logD)→ Rj∗CU is a quasi-isomorphism.

For example, Hk(U,C) ∼= Hk(Γ(V ,Ω•V (logD))).

Note: In particular, the topological Euler characteristic χ(U) of U may becomputed from Ω•V (logD). (As D is free, Ω1

V (logD) already carries thisinformation.)

This talk: A different instance of the same phenomenon.


D ⊂ V : hypersurface; U =complement of D.

Question: What should the ‘Chern class of U’ mean?

X compact, nonsingular ‘Chern class of X ’ = c(TX ) ∩ [X ].

What if X is singular? what if it is noncompact?

‘Chern class calculus’:

E.g., D nonsingular compact in V nonsingular compact. Then setcSM(U) := c(TV ) ∩ [V ]− c(TD) ∩ [D]. This is a class in V .

If X is nonsingular compact,cSM(X ) = c(TX ) ∩ [X ]− c(∅) = c(TX ) ∩ [X ].

Example, cSM(A3) =? A3 = P3 r P2, so

cSM(A3) = cSM(P3)− cSM(P2) = ((1 + h)4 − (1 + h)3h) ∩ [P3]

= [P3] + 3[P2] + 3[P1] + [P0] .


Theorem (MacPherson ’74, after a conjecture of Grothendieck-Deligne):

There is a unique theory of Chern classes for constructible functions:ϕ on X cSM(ϕ) ∈ H∗X (homology; or A∗X , Chow group)such that

X nonsingular compact =⇒ cSM(11X ) = c(TX ) ∩ [X ];

Homomorphism: cSM(ϕ+ ψ) = cSM(ϕ) + cSM(ψ);

Functoriality: f : X → Y proper, ϕ constructible on X=⇒ cSM(f∗ϕ) = f∗cSM(ϕ).

(Push-forward f∗ϕ: def’d by Euler characteristic of fibers.)

So: cSM is a natural transformation from the functor of constructiblefunctions to homology (Chow group. . . ).

Notation: If U ⊆ X , write cSM(U) for cSM(11U).


Example: f : X → point, U ⊆ X f∗11U = χ(U)11point.

Functoriality =⇒ f∗cSM(11U) = cSM(χ(U)11point) = χ(U) cSM(11point):∫cSM(U) = χ(U)

(‘Singular Poincare-Hopf’)

Also: if D1,D2 ⊆ X , then 11D1∪D2 = 11D1 + 11D2 − 11D1∩D2 . cSM(D1 ∪ D2) = cSM(D1) + cSM(D2)− cSM(D1 ∩ D2):‘inclusion-exclusion’.

In particular, cSM(U) = c(TV ) ∩ [V ]− c(TD) ∩ [D] if both D, V arenonsingular, compact.

So we do have a ‘Chern class calculus’.

Blanket assumption: V nonsingular compact; D ⊆ V divisor


Toy example: D nonsingular hypersurface in V nonsingular.D is free, and basic exact sequence

0 // DerV (− logD) // DerV // OD(D) // OJD(D) // 0

becomes

0 // DerV (− logD) // DerV // OD(D) // 0 .

Therefore

c(DerV (− logD)) =c(DerV )

c(OD(D))=

c(TV )

1 + D

= c(TV )− c(TV ) · D

1 + D

c(DerV (− logD)) ∩ [V ] = c(TV ) ∩ [V ]− c(TD) ∩ [D] .


Side-by-side:

cSM(V r D) = c(TV ) ∩ [V ]− c(TD) ∩ [D]

c(DerV (− logD)) ∩ [V ] = c(TV ) ∩ [V ]− c(TD) ∩ [D]

and hence:D ⊂ V both nonsingular =⇒ c(DerV (− logD)) ∩ [V ] = cSM(V r D) .

Conjecture (—)

V nonsingular compact, D free divisor in V , U = V r D =⇒

c(DerV (− logD)) ∩ [V ] = cSM(U) .

Straightforward: OK for simple normal crossings divisors. However:

dimV = 2, D = reduced curve with one singularity s.t. Milnor 6= Tyurina.Then D is free (depth JD = 2 = codim JD, so C-M);

but∫c(DerV (− logD)) ∩ [V ] 6= χ(U) =

∫cSM(U).


In fact:

Theorem (X. Liao, arXiv:1201.6110)

dimV = 2, D ⊂ V reduced curve, U = V r D.Then c(DerV (− logD)) ∩ [V ] = cSM(U) if and only if Milnor=Tyurina atall singularities of D.

Liao proves that c(DerV (− logD)) ∩ [V ] = cSM(U) if and only if

c(OJD(D)) ∩ [V ] = [V ]− s(JD,V )∨ .

How? Left-hand-side from basic exact sequence for DerV (− logD);

right-hand-side from a formula for cSM classes in terms of Segre classes(—, 1999).

If dimV = 2, then degree of l.h.s.=Tyurina, and degree of r.h.s.=Milnor.

Next obvious question: For what kind of singularities is Milnor=Tyurina?


Conjecture, revised

V nonsingular compact, D free, locally quasi homogeneous divisor in V ,U = V r D. Then c(DerV (− logD)) ∩ [V ] = cSM(U).

N.B.: Same hypothesis as for the logarithmic comparison theorem.Remarks:

Very different beasts: no immediate ‘functoriality’ on theleft-hand-side, while the right-hand-side is all about functoriality;

MacPherson’s definition of cSM involves interesting invariants ofsingularities (local Euler obstruction). Role of such invariants inDerV (− logD)?

Right-hand-side is often computable in practice.

Another instance of ‘access to complement’ principle.


Conjecture

V nonsingular compact, D free, locally quasi homogeneous divisor in V ,U = V r D. Then c(DerV (− logD)) ∩ [V ] = cSM(U).

Strongest ‘on the nose’ result I know:D is a hypersurface arrangement in V if D is a union of nonsingularhypersurfaces, locally analytically isomorphic to a hyperplane arrangement.

Theorem (—, arXiv:1201.5396)

If D is a free hypersurface arrangement in V , and U = V r D, thenc(DerV (− logD)) ∩ [V ] = cSM(U).

In particular:

Theorem

If A is a free hyperplane arrangement in Pn, thenc(DerPn(− logA)) ∩ [Pn] = cSM(Pn r A).

I will sketch of proof of particular case (from arXiv:1103.2777). Notrepresentative of general argument, but pretty story.


But first. . .

Theorem (X. Liao, arXiv:1205.3843)

(Any dimension!) The conjecture is true ‘numerically’, i.e.:If ι : V → Pn is an embedding, then

ι∗c(DerV (− logD)) ∩ [V ] = ι∗cSM(U) .

if D locally quasi homogeneous free divisor of V .

Corollary

The conjecture is true for V = Pn!


Key facts:under free quasi homogeneous hypothesis,∫

c(DerV (− logD)) ∩ [V ]) = χ(U) =∫cSM(U)

(logarithmic comparison theorem)

Both sides have same behavior under general hyperplane sections.

Behavior of cSM under hyperplane sections:‘same’ as nonsingular case (old story).Simple matter, but with interesting consequences(Dimca-Papadima-Huh formulae, Dolgachev’s conjecture. . . )

Behavior of c(DerV (− logD)) ∩ [V ]: see Liao’s paper.

After Liao’s theorem, the conjecture could only fail to be true on the nose(i.e., in the Chow group) only for very pathological reasons. . .But a general proof in the Chow group is not yet available.


Example: free hyperplane arrangements.

—I will sketch one proof below—Particular case of free hypersurface arrangement case—Now also implied by Liao’s theorem.

A hyperplane arrangement is a collection of distinct hyperplanes in Pn;equivalent information: corresponding affine, central arrangement.

A

PI2

O

A

A3

A


A little combinatorics:

A L(A), the poset of intersections, ordered by reverse inclusion µ(x , y), Mobius function χ

A(t), characteristic polynomial.

Mobius: µ(x , x) = 1;∑

x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :






x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :

−1

V

O

−1






x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :

1

V

O

−1 −1






x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :

1

V

O

−1 −1 −1 −1






x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :

1

V

O

−1

1

−1 −1 −1






x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :

1

V

O

−1

1

−1 −1 −1

11






x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :

2

V

O

−1

1

−1 −1 −1

1 1 1






x≤z≤y µ(x , z) = 0 for x < y .

E.g., µ(0, x) :

−2

V

O

−1

1

−1 −1 −1

1 1 12


Characteristic polynomial: χA

(t) =∑

x µ(0, x) tdim x

dim=3

−1

1

−1 −1 −1

1 1 12

−2

1

−4

5

−2dim=0

dim=1

dim=2

Here: t3 − 4t2 + 5t − 2= (t − 1)(t2 − 3t + 2). In general, χA

(1) = 0.

Poincare polynomial: πA

(t) = (−t)n+1 · χA

(−t−1)

Here: 1 + 4t + 5t2 + 2t3 = (1 + t)(1 + 3t + 2t2).



(t) =∑

x µ(0, x) tdim x

−2

−1

1

−1 −1 −1

1 1 12

−2

dim=3

dim=2

dim=1

dim=0

1

−4

5

Here: t3 − 4t2 + 5t − 2= (t − 1)(t2 − 3t + 2). In general, χA

(1) = 0.

Poincare polynomial: πA

(t) = (−t)n+1 · χA

(−t−1)

Here: 1 + 4t + 5t2 + 2t3 = (1 + t)(1 + 3t + 2t2).


Small variations:

χA

(t) :=χA

(t)

t − 1, π

A(t) :=

πA

(t)

1 + t

In the example:

χA

(t) = t2 − 3t + 2 , πA

(t) = 1 + 3t + 2t2

Context:

Theorem (Orlik-Solomon ’80)

πA

(t) =∑n

k=0 rkHk(Pn r A,Q)tk


[Pn r A] = χA

(L) in the Grothendieck group of varieties.

Here L is the class of the affine line in K (Var).


K (Var): free abelian group on iso classes of varieties, modulo relations

Basic relation: if Z ⊆ X closed embedding, then [X ] = [Z ] + [X r Z ].

As a ring: set [X ] · [Y ] = [X × Y ].

. . . important in motivic integration, stable birational geometry, ‘Hirzebruchclasses’ of varieties, ‘polynomial countability’ of graph hypersurfaces, . . .

Universal Euler characteristic: Every invariant of alg. varieties that isadditive on disjoint unions and multiplicative on products factorsthrough K (Var).In particular: the Hodge-Deligne polynomial (keeping track of ranks inmixed Hodge structure) may be computed from the Grothendieck class[X ] ∈ K (Var).

Example: L = [A1], Hodge polynomial: uv .[Pn] = 1 + L + · · ·+ Ln: Hodge polynomial = 1 + uv + · · ·+ unvn.



(t) =∑

x µ(0, x) tdim x


[Pn r A] = χA

(L) in the Grothendieck group of varieties.

This result is essentially trivial, and it implies Orlik-Solomon! (modulopurity of mixed Hodge structure of Pn r A)


Corollary

cSM(Pn r A) = χA

(t + 1)

where tk ↔ [Pk ].

This is also easy to prove independently, by using inclusion-exclusion.

Now, “Two months in the lab can save two hours in the library”:

Theorem (Mustata-Schenck ’01)

For free hyperplane arrangements, c(Ω1Pn(logA)⊗ OPn(1)) = π

A(h).

Undo twisting, dualize, sort through notation, getc(DerPn(− logA)) ∩ [Pn] = cSM(Pn r A),i.e., this case of the conjecture.

Details in arXiv:1103.2777.


Moral:

Maybe cSM(V r D) should be considered as the most naturalgeneralization of the characteristic polynomial from hyperplanearrangements to arbitrary divisors.

For free hyperplane arrangements,

c(Der(− logD)) ∩ [V ] oo // characteristic polynomial oo // cSM(V r D)

For more general free divisors? Even for free hypersurface arrangements,

c(Der(− logD)) ∩ [V ] oo? // characteristic polynomial oo

? // cSM(V r D)

Proof in this case: completely different. (See arXiv:1201.5396.)


Idea: Prove that c(DerV (− logD)) ∩ [V ] is ‘functorial enough’ to force itto agree with cSM(V r D).

What is ‘enough’?

S ⊆ D: minimal nonempty among intersections of components of D.(Hypothesis on D =⇒ S nonsingular.)

π : V → V : blow-up of V along S ;D ′: obtained from proper transforms of D, plus exceptional divisor.(Hypothesis on D =⇒ D ′ is also a free hypersurface arrangement.)

Note that π is iso (V r D ′)→ (V r D).

Functoriality of cSM =⇒ π∗cSM(V r D ′) = cSM(V r D).

Key: Enough to show that

π∗(c(DerV (− logD ′)) ∩ [V ]) = c(DerV (− logD)) ∩ [V ] .


π∗(c(DerV (− logD ′)) ∩ [V ]) = c(DerV (− logD)) ∩ [V ] (*)

Reason: May resolve singularities of D by a sequence of blow-ups at suchcenters.

So if (*) holds, then reduced to case of normal crossing divisors, whereconjecture is true.

How to prove (*)?

Approach in arXiv:1201.5396: use MacPherson’s graph construction toevaluate correction term(

c(DerV (− logD ′))− π∗c(DerV (− logD)))∩ [V ]

Here use the fact that both Der’s are locally free.Then prove that push forward of correction term vanishes, by a dimensioncount.


This is very technical. Better approach?

Almost true: It would be enough to show that with π as above,

π∗(c(π∗OJD(D)) ∩ [V ]) = c(OJD(D)) ∩ [V ] . (**)

(Actually, must use a slightly different JD here.)

Such ‘projection formulas’ are automatic for locally free sheaves, but donot hold for arbitrary coherent sheaves: need contributions from higherTor’s to vanish after push-forward.

Remarkable here: projection formula does hold in this case, by otherapproach. It would be interesting to understand better ‘why’ it does.


Thanks for your attention!


Seville|July 18th, 2012arXiv:1207.4202 X. Liao:Chern classes of logarithmic vector elds...

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