About the Calabi-Yau theorem and itsapplications

– Krageomp program –

Julien KellerL.A.T.P., Marseille University – Universite de Provence

October 21, 2009

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Any comments, remarks and suggestions are welcome. These notes aresumming up the series of lectures given by the author during the Krageomp

program (2009). Please feel free to contact the author atjkeller@cmi.univ-mrs.fr

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1 About complex and Kahler manifolds

In this lecture, starting with the background of Riemannian geometry, weintroduce the notion of complex manifolds:

• Definition using charts and holomorphic transition functions

• Almost complex structure, complex structure and Newlander-Nirenbergtheorem

• Examples: the projective space , S2, hypersurfaces, weighted projectivespaces

• Uniformization theorem for Riemann surfaces, Chow’s theorem (com-plex compact analytic submanifolds of CPn are actually algebraic vari-eties).

We introduce the language of tensors on complex manifolds, and some naturalinvariants of the complex structure:

• k-forms, exterior diffentiation operator, d = ∂+ ∂ decomposition, (p, q)forms

• De Rham cohomology, Dolbeault cohomology, Hodge numbers, Eulercharacteristic invariant

Using all the previous material, we can explain what is a Kahler manifold.In particular we will formulate the Calabi conjecture on that space.

• Kahler forms, Kahler classes, Kahler cone

• The Fubini-Study metric on CPn is a Kahler metric

Finally, we sketched the notion of holomorphic line bundle, the correspon-dence with divisor and Chern classes.

This very classical material can be read in different books:

- F. Zheng, Complex Differential Geometry, Studies in Adv. Maths,AMS (2000)

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- W. Ballmann, Lectures on Kahler manifolds, Lectures in Math andPhysics, EMS (2006). Seehttp://www.math.uni-bonn.de/people/hwbllmnn/notes.html

- J.P. Demailly, Complex analytic and algebraic geometry,http://www-fourier.ujf-grenoble.fr/~demailly/books.html

- V. Bouchard, Lectures on complex geometry, Calabi-Yau manifolds andtoric geometry, http://arxiv.org/abs/hep-th/0702063

- A. Moroianu, Kahler geometry, see also his recent book (London Math-ematical Society Student Texts 69, Cambridge University Press, Cam-bridge, 2007)http://www.math.polytechnique.fr/cmat/moroianu/publi.html

- D. Huybrechts, Complex geometry : an introduction, Springer (2004).Universitext.

- C. Voisin, Theorie de Hodge et geometrie algebrique complexe , Coursspecialises 10, SMF 2002. Translation in english: Hodge Theory andcomplex algebraic geometry I and II, Cambridge University Press 2002-3.

- P. Griffiths and J. Harris, Principles of Algebraic geometry, Wiley-Interscience (1994)

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2 The Calabi conjecture and Donaldson’s ap-

proach

2.1 What is a Calabi-Yau manifold ?

We start our lecture by giving four1 different definitions of a Calabi-Yaumanifold (M,J, ω): this is a compact Kahler2 manifold such that one of thefollowing conditions is satisfied

• Ric(ω) = 0, i.e there is a Ricci flat metric,

• c1(M) = 0,

• the canonical line bundle KM = ΛnT ∗M is trivial, where n = dimCM ,

• there exists a nowhere vanishing holomorphic n-form.

Note that here the crucial point is to show that any of the last 3 conditionsimply the existence of a Ricci flat metric. This is the famous Calabi conjec-ture (1954) solved by S.T. Yau (1978). This can be rephrased in terms ofcomplex Monge-Ampere equation and it turns out that one needs to showthat if ν is a smooth volume form and [ω] a given Kahler class, then thereexists a unique smooth potential φ solution of the highly non linear PDE:

(ω +√−1∂∂φ)n = ν.

The existence of φ is proved by a continuity method in Yau’s paper. Oneconsiders the 1-parameter family (0 ≤ t ≤ 1)

(ω +√−1∂∂φ)n = etf+ctωn

where Ric(ω) =√−1∂∂f and ct is a constant such that

∫Metf+ctωn = [ω]n.

Of course there is a solution at t = 0 (φ0 = 0) and we denote by S ⊂ [0, 1]the set of t for which there do exist solutions to this equation. Yau proved

1We will not need to discuss about holonomy but there is also a way to characterizeCalabi-Yau manifolds thanks to their holonomy.

2For symplectic or complex non Kahler manifolds, the interested reader will findsome engrossing recent achievements in http://arxiv.org/abs/0802.3648 (J. Fine andD. Panov), or http://arxiv.org/abs/math/0205012 (J. Gutowski, S. Ivanov, G. Pa-padopoulos).

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that S is open (using the implicit function theorem) and closed (this is thehard part, especially the C0 estimate). For the openesse, the ellipticity of theLaplacian of the ambiant metric is crucial, while the closedness is based onSchauder theory and clever a priori estimates. Improvements of this methodhave been achieved by S. Kolodziej and Z. Blocki for degenerate right handside using pluripotential theory. On another hand, H.D. Cao proved theCalabi conjecture using Kahler Ricci flow.

We give examples of Calabi-Yau manifolds by using a computation of thefirst Chern class of a hypersurface of degree d in CPn. Unfortunately, we don’thave time to discuss applications to topology (Miyaoka-Yau’s inequality, Fanomanifolds are simply connected), to rigidity of the Calabi-Yau theorem (seeF. Zheng’s book, supra reference) or Mirror symmetry.

2.2 Balanced metric

Consider now a projective Calabi-Yau manifold M of complex dimension n,and we will work with integral Kahler classes. Let us fix an ample line bundleL, assume that Lk is very ample3 and ν a smooth4 volume form. We willdenote Met(Ξ) the space of smooth hermitian metrics on the vector space orvector bundle Ξ. Then we have

Theorem 1. Let us consider h ∈Met(L) a smooth hermitian metric over Lwith positive curvature c1(h) > 0. Then we can define the Bergman functionover M

Bk(p) =

dimH0(Lk)∑i=1

|Si|2hk(p)

where Si is an orthonormal basis of H0(Lk) with respect to∫Mhk(., .)ν and

p ∈M . We have an asymptotic when k tends to infinity:

Bk(p) = knc1(h)n

ν+ kn

c1(h)n

ν

scal(c1(h))

2+O(kn−2).

3In the language of physicists we have a positive polarisation, i.e we are in the contextof geometric quantization.

4Actually, one could choose generalize to some extent our results in the non smoothcase.

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The Bergman function is the restriction over the diagonal of the integralkernel of the L2-projection from the space of smooth sections of Lk to thespace of holomorphic sections of Lk. The asymptotic formula can be inter-preted as a Riemmann-Roch pointwise formula and the first two terms aresimple geometric quantities (scal(ω) stands for the scalar curvature of themetric ω that is the trace of the Ricci curvature Ric(ω)). The proof can beobtained by updating a proof of G. Tian from his paper “On a set of po-larized Kahler metrics on algebraic manifolds” (JDG, 1990)5 by using peaksections. Noticing that the Bergman function is depending on the choice ofthe metric on the fiber, S. Donaldson defined the notion of ν-balanced metric

Definition 1. A metric h ∈Met(L)k is ν-balanced or order k if the functionBk,h is constant over the manifold, that is

Bk,h(p) =dimH0(Lk)

V olν(M)

for all p ∈M .

Then, we can introduce two natural maps:

• FS : Met(H0(Lk))→Met(Lk) such that for all p ∈M ,

dimH0(Lk)∑i=1

|Si|2FS(H)(p) =dimH0(Lk)

V olν(M)

where Si is H-orthonormal basis.

• Hilbν : Met(Lk)→Met(H0(Lk)) such that for any h ∈Met(Lk),

Hilbν(h) =

∫M

h(., .)ν,

is just the natural induced L2 metric induced by h, ν.

Note that if h is ν-balanced, then we will say that H = Hilbν(h) is ν-balancedand we get for such a metric∫

M

< Si, Sj >FS(H) ν = δi,j

5See also T. Bouche’s paper in references.

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for (Si) an H-orthonormal basis. Now, we have a natural dynamical sys-tem induced by the composition of the maps Hilbν and FS on the space ofmetrics. Donaldson showed that it has a trivial behaviour:

Theorem 2. Let ν any given smooth volume form. Then the iterations ofHilbν FS give a dynamical system with fixed attractive points, unique upto action of U(H0(Lk)). Balanced metrics are limits of the iterations of theHilbν FS map.

The proof is very natural. If we set ψz(H) = 1N

log |z|2H + 1N

log detH,then we can consider the functional over Met(H0(Lk)),

Ψν(H) =

∫M

ψz(H)ν

Actually Ψν is convex and proper. It is the finite dimensional analog of theso-called Aubin-Yau functional. Its critical points are precisely ν-balancedmetrics. Finally, using the arithmetico-geometric inequality and the concav-ity of the log, one can check that an iteration of Hilbν FS decreases thefunctional Ψν .

So we get for all k large enough a ν balanced metric in a canonical way.These metrics are algebraic, i.e appear as Fubini-Study type metrics for cer-tain embeddings M → PH0(Lk)∨, while the Calabi-Yau metric is a tran-scendental object (except in some very rare simple cases, see for instance D.Hulin Sous-varietes complexes d’Einstein de l’espace projectif, http://www.numdam.org/numdam-bin/fitem?id=BSMF_1996__124_2_277_0). A key is-sue is now to show that the sequence of ν-balanced metrics converge when ktends to infinity. From Theorem 1, we know that if it is convergent, then atinfinity we get a metric h∞ such that

(c1(h∞))n = ν

i.e we have solved the Calabi conjecture, by prescribing in a given Kahlerclass c1(L) the volume form of the Kahler metric. In order to show theconvergence, we can use Yau’s theorem (so this method does not give analternative proof for the moment). This is quite complicated and use a subtileimplicit type function theorem based on the structure of double symplecticquotient (explained later, with the quantification of the Hermitian-Einsteinequation).

Some references:

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- Z. Blocki, The complex Monge-Ampere equation on compact Kahlermanifolds, Course given at the Winter School in Complex Analysis,Toulouse, http://gamma.im.uj.edu.pl/~blocki/publ/ln/index.html

- S. Kolodziej, The complex Monge-Ampere equation and pluripotentialtheory. Mem. Amer. Math. Soc. 178 (2005)

- J.P. Bourguignon, Premieres formes de Chern des varietes Kahleriennescompactes, Seminaire Bourbaki 507., Lecture notes in mathematics 710(1977-1978)

- Y.T. Siu, Lectures on Hermitian-Einstein metrics for stable bundlesand Kahler-Einstein metrics, Birkhauser, (1987)

- T. Aubin, Nonlinear Analysis on Manifolds, Monge–Ampere Equations.Springer (1998)

- S. Donaldson, Some numerical results in complex differential geometry(2005), http://arxiv.org/abs/math/0512625

- S. Donaldson, Scalar curvature and projective embeddings, II QuarterlyJour. Math. 56 (2005), http://arxiv.org/abs/math/0407534

- J. Keller, Ricci iterations on Kahler classes, available on my website,(2007), http://www.cmi.univ-mrs.fr/~jkeller/Julien-KELLER-publi.html

- T. Bouche, Convergence de la metrique de Fubini-Study d’un fibrelineaire positif, Ann. Inst. Fourier (1990).http://www.numdam.org/numdam-bin/fitem?id=AIF_1990__40_1_117_

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- X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergmankernels, Birkhauser, 2007.

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3 Numerical approximations of Calabi-Yau met-

rics

We have seen in the previous lecture a canonical way to find a ν-balancedmetric and that ν-balanced metrics of order k converge when k tends toinfinity towards the solution to the Calabi conjecture. Actually, a moreprecise statement is that one has convergence with quadratic speed in k.Thus we obtain an algorithm in order to compute a Calabi-Yau metric in anintegral Kahler class:

1. Found a large number of points over the manifolds.

2. Fix the volume form. Compute the volume at the points chosen previ-ously

3. Fix the space of holomorphic sections H0(Lk). Use the symmetries toreduce the dimension if possible

4. Fix a random hermitian matrix H0 ∈Met(H0(Lk))

5. Compute the iterates (Hilbν FS)r(H0) till one converges (usuallyr ∼ 10 is sufficient). This requires to know the points over the manifoldand inverse a matrix to get an orthonormal basis.

The algorithm for finding the ν-balanced metric has an exponential speed ofconvergence dependant on the smallest positive eigenvalue of the Laplacianof the Calabi-Yau metric A key issue here is also the complexity of the algo-rithm. For a complex n dimensional manifold, this turns out to be equivalentto k4n, where most of the computations are done to evaluate the Bergmanfunction (essentially a “polynomial” of degree k in n variables) over the wholeset of points representing the manifold. In particular it is clear that using thismethod, one can compute with a single desktop computer any Calabi-Yaumetric on complex surfaces (even with no symmetry).

Some examples

We discuss the basic example of CP1 and the Fubini-Study metric. In thatcase the ν-balanced metric is precisely the solution to the Calabi conjecture(we have fixed the volume form by chosing precisely the Fubini-Study metric).

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A more interesting case is given by a double cover of the plane branchedover the sextic curve x6 +y6 +z6 = 0. We will denote by S this K3 surface. Itwas studied by S. Donaldson who obtained a formula to compute its volume.Here the nowhere vanishing 3-form is given by dx∧dy

win the chart w2 =

x6 + z6 + 1. Over CP2, the map

f : [x, y, z] 7→ [x3, y3, z3],

maps the sextic branch curve x6 +z6 +z6 = 0 to the conic X2 +Y 2 +Z2 =0 and preimages of f are elliptic curves C = p3 + q3 + 1 = 0. Thedouble cover S → CP2, is the pull-back by f of the covering of the quadric∼ CP1 × CP1 over the plane branched along the conic X2 + Y 2 + Z2 =0. Thus, one get a holomorphic isomorphism that sends an open subset ofCP1 × C to an open dense subset of S. By using the elliptic fibration andthe symmetries, one obtains explicitly 2 charts that constitue almost an atlasof the manifold. This allows us to get the points in the computer program,and at the same time to compute the theoretical value of the volume of themanifold with respect to the chosen volume form.

The results show a good accuracy of the whole algorithm. It is also verystable (if we modify the volume a little bit, one does not change much theν-balanced metric, which can be interpreted as a form of stability of theMonge-Ampere, see S. Kolodziej). Note that if one linearizes the Hilbν FSmap, one gets around the balanced point that its differential is given by

ε 7→ 1

dimH0(Lk)

∫M

∑γ,δ

< sα, sβ >< sγ, sδ > εγ,δν

Using peak sections, one can show that this operator is the quantification ofthe operator

e−∆ω∞4πk

where ω∞ is the Ricci flat metric. Hence, as a by product, it is possibleto obtain the spectrum of the Laplacian of the Ricci flat metric. This alsoprovides a Newton type method close to the balanced point that gives arefinement of the original method with a better precision.

An even more interesting example to study, is the family of Fermat quin-tics in CP4 :

z50 + z5

1 + z52 + z5

3 + z54 − 5ψz0z1z2z3z4 = 0

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Then, one can estimate the number of holomorphic sections for O(k). Onechooses for the following 3-form using the Poincare residue map,

Ω =ψdZ0 ∧ dZ1 ∧ dZ2

Z43 − ψZ0Z1Z2

in affine coordinates. This is a natural choice coming from physics consider-ations.

In order to find points over the manifolds, one can use a Monte-Carlotype method. We give some explanations about this idea now. Recall thatfor a random variable Y on the probability space (Γ, dγ), the expected valueof Y in dγ is E(Y ) =

∫ΓY dγ. Consider S = s ∈ H0(Lk0) : ‖s‖L2 = 1 the

unit sphere that we can equip with the Fubini-Study Kahler form. Then, letus choose s ∈ H0(Lk0), and define

T : s 7→ Ts =1

2π

√−1∂∂ log |s|2

which is the current of integration given by Lelong-Poincare formula. Alemma of Zelditch and Shiffman (see http://arxiv.org/abs/math/9803052,section 3) asserts that the expected value

E(T ) =

∫STsdµFS(s)

is smooth for any k0 for which Lk0 is very ample. Furthermore,

E(T ) =1

kι∗k0

(ωFS,PH0(Lk0 )∨)

where ιk0 is the embedding of the manifold into the projective space PH0(Lk0)∨.The same applies to current associated with the simultaneously vanishing ofp ≥ 1 random section of S. Thus, in order to find points on the manifolds, itis sufficient to intersect the manifolds with lines6 (in the smallest projectivespace in which one can embed the manifold, i.e the smallest k0 > 1), get theintersection points, and one knows now that they are distributed accordingto the Fubini-Study volume form induced by the embedding of the manifold.This turns out to be a very efficient method together with a Newton-Raphsonalgorithm that enables to find roots of a polynomial of one variable. Thus

6a line is seen as the intersection of n random hyperplanes.

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one obtains the points over the manifold and it is possible to do numericalintegration over the quintic.

Most of the program presented during my talk can be downloaded frommy webpage:http://www.cmi.univ-mrs.fr/~jkeller/Julien-KELLER-progs.html.The precision of the computation of the Ricci flat metrics are in averagearound 1% in L1 norm (and for a few minutes of computations). The code iswritten in C++. Feel free to modify it to obtain Ricci flat metrics on othermanifolds !

Finally, always in the projective context, other algorithms have been de-veloped in order to find numerical solutions of Hermitian-Einstein equations(Yang-Mills equations) for vector bundles (and their generalizations, like Vor-tex equations), constant scalar curvature Kahler metrics (especially FanoKahler-Einstein toric metrics). This will be adressed in a forthcoming talk.

Some references :

- M. Douglas, R. Karp, S. Lukic, R. Reinbacher, Numerical Calabi-Yau metrics, J. Math Physics, 2008. http://arxiv.org/abs/hep-th/0612075

- V. Braun, T. Brelidze, M. Douglas, B. Ovrut, Calabi-Yau metrics forquotients and complete intersections, JHEP 2008.http://arxiv.org/abs/0712.3563

- M. Douglas, R. Karp, S. Lukic, R. Reinbacher, Numerical solution tothe hermitian Yang-Mills equation on the Fermat quintic, JHEP 2007.http://arxiv.org/abs/hep-th/0606261

- S. Donaldson, Some numerical results in complex differential geometry(2005), http://arxiv.org/abs/math/0512625

- P. Candelas, X. de la Ossa, P. Green, L. Parkes, A pair of Calabi-Yaumanifolds as an exactly soluble superconformal theory.

- S. Kolodziej, Stability of solutions to the complex Monge-Ampere equa-tion on compact Kahler manifolds, Indiana Univ. Math. J

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4 Quantification of the Hermitian-Einstein met-

ric

In this lecture, we are interested in the famous Kobayashi-Hitchin correspon-dance and in Hermitian-Einstein metrics. Given (M,ω) a Kahler manifold7

and E a holomorphic vector bundle on M , one says that the hermitian metrich ∈Met(E) on E is Hermitian-Einstein if it satisfies the following PDE:

√−1ΛωFh = µ(E)IdE ∈ End(E)

where Λω is the dual of the Lefschetz operator L(u) = ω∧u. Note that µ(E)is here a topological constant, the slope of the bundle, given by

µ(E) = µL(E) =deg(E)

rank(E)=

∫Mc1(E) ∧ c1(L)n−1

(n−1)!

rank(E).

It is not true that any vector bundle E carries such a metric. On anotherhand, in the case of rank(E) = 1, the equation just involves the Laplacian.This means that in order to find a Hermitian-Einstein metric, one just needsto inverse the operator

√−1Λω∂∂ which is possible. Thus, there exists always

a Hermitian-Einstein metric on a line bundle.The Kobayashi-Hitchin correspondance relates the existence of Hermitian-Einstein metric to an algebraic condition, the Mumford-Takemoto stability.

Definition 2. Let E a holomorphic vector bundle. Then E is said to beMumford stable if for any coherent subsheaf F with rank(F) < rank(E), onehas

µ(F) < µ(E).

If the inequality is large, one speaks of Mumford semi-stability.

The Kobayashi-Hitchin correspondance proved by Narasimhan-Seshadri(in dimension 1), Donaldson (for projective manifolds), Uhlenbeck-Yau (forKahler manifolds), asserts that if E is an irreducible vector bundle, the exis-tence of a Hermitian-Einstein metric is equivalent to the Mumford stability ofE. This is a striking result in complex geometry since it relates a global alge-braic condition to a local intrinsic one. Note that it has various applications

7actually for the sake of clearness, we choose to work with ω in an integral class, thatit to see ω as the curvature of a hermitian metric on a polarisation L. Much of what weexplain in that section can be adapted to the Kahler non-projective case.

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in topology, or in algebraic geometry through the study of moduli spacesof solutions. For instance, such a correspondance is used in the work of A.Teleman to complete the classification of class V II0 (non-algebraic) complexsurfaces with b2 = 1 (Cf. Nakamura and Kato’s conjecture of global sphericalshell for the VII class). For general introduction on the Kobayashi-Hitchincorrespondance, we recommend to read:

- Lubke, Martin and Teleman, Andrei, The Kobayashi-Hitchin correspon-dence, World Scientific Publishing Co. Inc. (1995).

- Huybrechts, Daniel and Lehn, Manfred, The geometry of moduli spacesof sheaves, Aspects of Mathematics, E31 (1993)

- Donaldson, S. K. and Kronheimer, P. B., The geometry of four-manifolds,Oxford Science Publications (1990)

- S. Kobayashi, Differential geometry of complex vector bundles, Publ.Math. Soc. of Japan, Princeton Univ. Press (1987)

On another hand, another notion is very used in that set up and enablesus to construct quasi-projective moduli spaces of vector bundles over a pro-jective manifold (M,L). This is the notion of Gieseker-Maruyama stabilitythat we present now. First of all, we recall the definition of the Hilbert poly-nomial for a sheaf F , i.e χF(k) = χ(F ⊗Lk) =

∑i(−1)i dimH i(M,F ⊗Lk).

Remark that by the ampleness of L, this sum reduces to the dimension ofthe space of holomorphic sections for k large enough.

Definition 3. Let E a holomorphic vector bundle. Then E is said to beGieseker stable if for any coherent subsheaf F with rank(F) < rank(E), onehas

χ(F ⊗ Lk)rank(F )

<χ(E ⊗ Lk)rank(E)

,

for k >> 0. If the inequality is large one speaks of Mumford semi-stability.A direct sum of Gieseker stable bundles is called Gieseker polystable. Finallyif E ⊗ Lk satifies the previous inequality, we will say that E is k-Giesekerstable (resp. semistable or polystable).

Note that the Mumford stability implies Gieseker stability. Building fromGieseker’s construction, X. Wang proved that the Gieseker poystability ofthe bundle E of rank r = rank(E) is equivalent to the G.I.T stability of

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the corresponding point [E] ∈ PHom(ΛrH0(M,E ⊗ Lk), H0(det(E ⊗ Lk))∗under the action of SL(H0(M,E⊗Lk)). This point is naturally given by anembedding of the manifold in the Grassmannian Gr(r,N) of r-planes in CN

where N = dimH0(E ⊗ Lk) using global holomorphic sections of E ⊗ Lk.Using the natural symplectic form on C∞(M,Gr(r,N)), it is possible

to rephrase this result by saying that there exists an embedding ι : M →Gr(r,N) by holomorphic sections of E⊗Lk which is a zero of the moment mapµSU(N) corresponding to the natural action of G = SU(N) (GC = SL(N))on C∞(M,Gr(r,N)). By moment map, we mean the following.

Definition 4. Let (Y, χ) be a symplectic manifold, G a compact Lie groupacting on Y by symplectomorphisms (the action preserves the symplecticform). Then, a moment map for the action of G is a map µ : Y → Lie(G)∗

such that:

• if we set µX = 〈µ,X〉 (with 〈, 〉 the dual pairing on Lie(G)) is a Hamil-

tonian function, i.e if−→X (p) = ∂ exp(tX)(p)

∂t |t=0 is the vector field gen-

erated by X, thend〈µ,X〉(p) = ι−→

Xωp(., .),

where ι is the interior product (contraction of the form with the vectorfield).

• µ is required to be G-equivariant, where G acts on Lie(G)∗ via thecoadjoint action. Thus one obtains a homomorphism of Lie algebra.

Let us give an example. If one considers Cn with the standard formωstd =

√−12dzi ∧ dzi, then U(n) acts in an obvious way. For X ∈ Lie(U(n)),

one has−→X (z) = X · z. We check that if we set 〈µ,X〉(z) =

√−12z∗Xz, then

one gets a moment map:

d〈µ,X〉(z)(v) =

√−1

2(v∗Xz + z∗Xv)

=

√−1

2(v∗Xz − v∗Xz)

= −Im(v∗Xz)

= ωstd(X(z), v).

If we use the coupling (A,B) = −tr(AB) on Lie(U(N)) = A ∈ Mn(C) :

A + A∗ = 0, then we can identify the moment map with µ(z) = −√−12zz∗.

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Finally, a direct computation shows the U(N)-equivariance:

〈µ(z), [X1, X2]〉 = −Im(z∗X∗2X1z) = 〈µ,X1〉, 〈µ,X2〉(z),

so we have got a moment map as claimed.Using this computation, one sees immediately the moment map associated

to the SU(N) action on C∞(M,Gr(r,N)):∫M

(Z(p)Z∗(p)− r

NIdN

) ωn(p)

n!

where Z(p) is the Stiefel point in the Grassmannian (so that Z∗Z = Idr).To find a zero of this moment map is equivalent to find a metric H ∈Met(H0(E⊗Lk)) such that for (Si) an H-orthonormal basis of H0(E⊗Lk),one has ∫

M

< Si, Sj >ωn(p)

n!= δij,

where <,> denotes the Fubini-Study metric induced on E ⊗ Lk by H. Onanother words, it means that the pull-back by ι of the Fubini-Study metricfrom the Universal vector bundle (over Gr(r,N)) can be identified with themetric on E. Thus, there is a natural definition of balanced metric for vectorbundles that correspond to Gieseker polystability.

Definition 5. A metric h ∈Met(E⊗Lk) is said to be k-balanced if for given(Si) an orthonormal basis of H0(E⊗Lk) with respect to

∫Mh(., .)ω

n

n!, one has

dimH0(E⊗Lk)∑i=1

Si ⊗ S∗hi =N

rV ol(M)IdE.

A metric H ∈ Met(H0(E ⊗ Lk)) is said to be k-balanced if for (Si) an H-orthonormal basis of H0(E ⊗ Lk), one has ∀i, j,∫

M

< Si, Sj >ωn(p)

n!= δij,

where <,> is the Fubini-Study metric given by∑N

i=1 Si < ., Si >= NrV ol(M)

IdE,

with N = dimH0(E ⊗ Lk).

Theorem 3. The vector bundle E is Gieseker polystable if and only if forall k >> 0, there exists a k-balanced metric.

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We use these metrics to give a quantification of a Hermitian-Einstein met-ric that exists a priori. We wish to construct a sequence of balanced metricswhen one assumes the existence of a Hermitian-Einstein one. For that, weremark that balanced metrics appear as zeros of two moment maps and thatour problem can we sumed up in finding a point in a double symplecticquotient.

Consider the set

H = (s1, ..., sN , A) ∈ C∞(M,E ⊗ Lk)N ⊗A1,1(E) s.t. ∂Asi = 0.

On that space acts diagonally the gauge group G(E) = U(E) and SU(N).Actually, there is a symplectic form on that space (not necessarily smoothbut we just consider a complex orbit) such that the moment map associatedto the Gauge group action is precisely at pH = (s1, ..., sN , A) ∈ H,

µG(pH) =∑i

sis∗i .

For the SU(N)-action there is also a moment map and the zero of this mo-ment map correspond to orthonormal basis (s1, ..., sN) which are actuallyholomorphic sections. Thus, a point in the symplectic quotient H//(G ×SU(N)) correspond precisely to a k-balanced metric. In order to find sucha point, one needs first to find a point in H//G and then look for an or-thonormal basis. This first step is done by deforming the Hermitian-Einsteinmetric. This is a consequence of the asymptotic expansion for the Bergmankernel (generalization of Tian-Bouche’s result).

Proposition 4.1. Assume hE ∈ Met(E), hL ∈ Met(L) with positive cur-vature. Then if (Si) is an orthonormal basis of H0(E ⊗ Lk) with respect to∫MhE ⊗ hkL(., .)ω

n

n!and

B(hE, hL) =N∑i=1

Si ⊗ S∗hE⊗h

kL

i .

Then when k tends to infinity,

B(hE, hL) = knIdE + kn−1

(√−1ΛωFhE +

1

2scal(ω)IdE

)+O(kn−2)

A point in H//G is given by the following result.

18

Proposition 4.2. Assume h is a Hermitian-Einstein metric on E. Thenfor r > 0, there exists a metric hk,r for k >> 0 such that

B(hk,r) = Cst× IdE +O

(1

kr

).

Then the metric hk,r ×(IdE +O

(1kr

))−1gives the expected point in

H//G, and we call it almost-balanced. Finally, in order to obtain a pointin (H//G)//SU(N), we just need in an SL(N)-orbit. Thanks to the nextresult due to Donaldson, we know that if the starting point is not too farfrom a zero of µSU(N), then the gradient flow will converge towards a zero,i.e a k-balanced metric.

Proposition 4.3. Let (Y, χ) be a symplectic manifold, G a compact Lie groupacting on Y by symplectomorphisms. Assume that µ is a moment map forthe G action on (Y, χ). Let σy : Lie(G) → TYy be the infinitesimal action.Assume that Qy = σyσ

∗y is invertible. Consider a point y0 ∈ Y such that

• |µ(y0)| < δλ

• ||Q−1y || < λ for all y = e

√−1θy0 avec |θ| < δ.

Then, there exists y1 with y1 = e√−1θ1y0, |θ1| < λ|µ(y0)| and µ(y1) = 0.

With Proposition 4.2, we can control the operator norm ||Q−1y || at the

almost-balanced metric hk,r. This use some classical L2 estimates. We don’tpresent details about it since it is conjectured that one can still improvetechnically this part. Finally, with Proposition 4.3, we have all the ingredientsto obtain

Theorem 4. Assume that E has a Hermitian-Einstein metric. Then thereexists a sequence of balanced metrics hk ∈ Met(E ⊗ Lk) such that hk ⊗ h−kLconverges to a metric h∞. Up to a conformal change, the metric h∞ ∈Met(E) is Hermitian-Einstein8.

An application of this theorem can be given for E = L a fixed line bundle.It shows that the algorithm presented at the end of Section 2.2 is convergent:the sequence of ν-balanced metrics is convergent towards the solution of the

8Actually h∞ is just almost-Hermitian Einstein, that is ΛFh∞ + 12scal(ω)Id = cst× Id

19

Calabi problem. Others generalizations of Theorem 5.2 include also the caseof coupled Vortex equations.

Some references :

- Y.T. Siu, Lectures on Hermitian-Einstein metrics for stable bundlesand Kahler-Einstein metrics, Birkhauser, (1987)

- S. Donaldson, Geometry in Oxford c. 1980–85, Asian J. Math. 3 (1999)

- S. Donaldson, Scalar curvature and projective embeddings, I Journ.Diff Geom (2001),

- J. Keller, Ricci iterations on Kahler classes, available on my website,Journ. Math Jussieu (2007),http://www.cmi.univ-mrs.fr/~jkeller/Julien-KELLER-publi.html

- J. Keller, Canonical metrics for Vortex type equations, Math. Annalen(2007)

- X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergmankernels, Birkhauser, 2007.

- Mumford, D. and Fogarty, J. and Kirwan, F., Geometric invarianttheory, Springer-Verlag (1994)

- X. Wang, Canonical metrics on stable vector bundles, Comm. Anal.Geom., 13 (2005).

- X. Wang, Balance point and stability of vector bundles over a projectivemanifold, Math. Res. Lett. 9 (2002)

20

5 Towards a new proof of the Calabi-Yau the-

orem

We present a natural way to solve the Calabi problem. This is a joint workwith H. D. Cao.

We have seen that if we set (M,L) a polarized manifold, ν a smoothvolume form, there exists for k >> 0 a notion of ν-balanced metric. Theyare fixed points of the operator FS Hilbν acting on Met(Lk). Another wayof presenting the notion of ν-balanced metric is to introduce a moment mapµ : CPNk → iu(N + 1) for the U(N + 1) action. Given homogeneous unitarycoordinates, one sets µ = (µ)α,β where

(µ)α,β[z0, ..., zN ] =zαzβ∑i |zi|2

.

Then, given an holomorphic embedding ι : M → PH0(Lk)∨, we can considerthe integral of µ over M with respect to the volume form:

µν(ι) =

∫M

µ(p)ν(p)

which provides a moment map for the U(N + 1) action over the space ofall bases of H0(Lk). Actually, there is a Kahler structure on that spaceisomorphic to GL(N + 1), and U(N + 1) acts isometrically with the momentmap which is essentially

−√−1

(µν −

tr(µν(ι))

N + 1IdN+1

).

The embedding ι is ν-balanced if and only if

µ0ν(ι) := µν(ι)−

tr(µν(ι))

N + 1IdN+1 = 0.

We know that a ν-balanced embedding corresponds (up to SU(N + 1)-isomorphisms) to an ν-balanced metric ι∗ωFS by pull-back of the Fubini-Study metric from PH0(Lk).

On another hand, seen as a hermitian matrix, µ0ν(ι) induces a vector field

on CPNk . Thus, inspired from the work of J. Fine, we study the followingflow

dι(t)

dt= −µ0

ν(ι(t)) (1)

21

and we call this flow the the ν-balancing flow. To fix the starting point ofthis flow, we choose a Kahler metric ω = ω(0) and we construct a sequence ofhermitian metrics hk(0) such that ωk(0) := c1(hk(0)) converges smoothly toω(0) providing a sequence of embeddings ιk(0) for k >> 0. Such a sequenceof embeddings is known to exist by a result of Tian. For technical reasons,we decide to rescale this flow by considering the following ODE.

dιk(t)

dt= −kµ0

ν(ιk(t)) (2)

that we call the rescaled ν-balancing flow. Of course, we are interested inthe behavior of the sequence of Kahler metrics ωk(t) = 1

kιk(t)

∗(ωFS) when tand k tends to infinity.

Theorem 5. For any t, the sequence ωk(t) converges in C∞ sense to thesolution ω +

√−1∂∂φt of the Monge-Ampere equation

∂φt∂t

= 1− ν

(ω +√−1∂∂φt)n

(3)

with φ0 = 0 and ω = limk→∞ ωk(0). We call this flow the ν-Kahler flow.

5.1 Study of the limit of the balancing flow

In that section, we assume that the sequence ωk(t) is convergent and wewant to relate its limit to Equation (3). Given a matrix H in Met(H0(Lk)),we obtain a vector field XH which induces a perturbation of any embeddingι : M → PH0(Lk)∨. The induced infinitesimal change in ι∗ωFS is pointwiselygiven by the potential tr(Hµ). Thus, the corresponding potential in the caseof the ν-balancing flow is

β = −tr(µ0νµ).

Since we are rescaling the flow in (2) and considering forms in the class2πc1(L), we are lead to understand the asymptotic behavior when k →∞ ofthe potentials βk = −ktr(µ0

νµ). Using Tian-Bouche asymptotic expansion,we can prove:

Proposition 5.2. Let hk ∈Met(Lk) be a sequence of metrics such that ωk :=1kc1(hk) is convergent in smooth topology to the Kahler form ω. Then the

22

potentials βk induced by the metrics Hilbk(hk) converge in smooth topologyto the potential

1− ν

ωn,

that is the potential of Equation (3).

Let us give a sketch of the proof. Let (si) be an orthonormal basis ofH0(Lk) with respect to the metric Hk := Hilbk(hk). The balancing potentialat p ∈M is

βk(Hk) = −∫M

∑α,β

(〈sα, sβ〉(q)∑Nki=1 |si(q)|2

− δαβNk + 1

)〈sα, sβ〉(p)∑Nki=1 |si(p)|2

νn(q),

where 〈., .〉 stands for the fibrewise metric hk. By Riemann-Roch theorem,Nk = knV ol(L) + O(kn−1). From Tian’s theorem and the fact that ωk isconvergent, we obtain

βk =Nk

Nk

− kn∑Nki=1 |si(p)|2

∫M

〈sα, sβ〉(q)〈sα, sβ〉(p)kn

(kn∑Nk

i=1 |si(q)|2ν

ωnk (q)

)ωnk (q)

= 1−(

1 +O

(1

k

))∫M

〈sα, sβ〉(q)〈sα, sβ〉(p)kn

((1 +O

(1

k

))ν

ωnk (q)

)ωnk (q)

But now, from Fine’s paper (Theorem 18), one knows the asymptotic behav-ior of the quantification operator

Qk(f)(p) =1

kn

∫M

∑α,β

〈sα, sβ〉(q)〈sα, sβ〉(p)f(q)ωnk (q).

Precisely, ‖Qk(f) − f‖Cm ≤ Ck‖f‖Cm for an independent constant C > 0.

Then, for k →∞,

βk(Hk)(p) = 1− ν

ωnkQk

(1 +O

(1

k

))One has convergence of Qk

(1 +O

(1k

))to 1 +O(1/k). This gives finally the

expected result.We are now ready for the main result of this section.

Theorem 6. Suppose that for each t ∈ [0, T ], the metric ωk(t) induced byEquation (2) converges in smooth topology to a metric ωt and, moreover, thatthis convergence is C1 in t. Then the limit ωt is a solution to the flow (3)starting at ω0 = limk→∞ ωk(0).

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5.3 Study of the ν-Kahler flow

We are now interested in the flow

∂φt∂t

= 1− ν

(ω +√−1∂∂φt)n

(4)

over a compact Kahler manifold (not necessarily in an integral Kahler class),where φ0 = 0 and ω is a Kahler form in the class [α]. Of course, this can berewritten as

(ω +√−1∂∂φt)

n =1

1− ∂φt∂t

efωn (5)

where f is a smooth (bounded) function defined by f = log(ν/ωn). Weare interested in the long time existence of this flow and its convergence.We remark that if we look at the formal level this equation in terms ofcohomology class, we obtain directly

∂[(ω +√−1∂∂φt)]

∂t= 0,

which shows that the Kahler form ω +√−1∂∂φt remains in the same class

as ω +√−1∂∂φ0, i.e. 2πc1(L). We study now long time existence and

convergence of this flow, following the ideas of Cao’s proof of the Calabiconjecture. For instance, by maximum principle, one gets:

Proposition 5.4. The function and ∂φt∂t

and 1

1− ∂φt∂t

remain bounded in C0

norm along the flow given by Equation (5).

We denote ∆ the Laplacian with respect to the Kahler form ω and by the operator ν

ωnt∆t− ∂

∂t. Applying the ideas of Yau and Cao (i.e the maximum

principle to ∆φ using the operator , one gets:

Lemma 5.1. One has a lower and upper bound for ∆φt. Then, there is ana priori C2 estimate for the ν-balancing flow.

Using these a priori estimates, one shows long time existence and conver-gence of the ν-balancing flow. Note that this flow has similar properties tothe J-flow studied by S.K. Donaldson, X.X. Chen, and B. Weinkove.

Some references :

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- H-D. Cao, Deformation of Kahler metrics to Kahler-Einstein metricson compact Kahler manifolds, Invent. Math, 81 (1985).

- H-D. Cao, J. Keller The Ω-balancing flow, Preprint 2009

- X.X. Chen and S. Sun, Space of Kahler metrics (V)— Kahler quanti-zation, Arxiv 0902.4149v1 (2009)

- S.K. Donaldson, Some numerical results in complex differential geom-etry, arXiv (2005).

- J. Fine, Calabi flow and projective embeddings, Arxiv 2009

- X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergmankernels, Progress in Mathematics, 254, Birkhauser Verlag, (2007)

- G. Tian, On a set of polarized Kahler metrics on algebraic manifolds,Jour. Diff. Geom 32 (1990)

- S.-T. Yau, On the Ricci curvature of a compact Kahler manifold and thecomplex Monge-Ampere equation I. Comm. pure appl. math, (1978).

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Forthcoming talks

• A variational approach to complex Monge-Ampere equations.

• Geodesics in the space of Kahler metrics

• About Fano manifolds. Tian’s α invariant, Nadel’s ideal sheaf, Demailly-Kollar theorem, and properness of the K-energy. Applications to thetoric case.

• Applications. Stability for projective manifolds and Yau-Tian-Donaldsonconjecture.

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