About the Calabi-Yau theorem and its applications

– Krageomp program –

Julien Keller L.A.T.P., Marseille University – Universit´ede Provence

October 21, 2009

1 Any comments, remarks and suggestions are welcome. These notes are summing up the series of lectures given by the author during the Krageomp program (2009). Please feel free to contact the author at [email protected]

2 1 About complex and K¨ahlermanifolds

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

• Definition using charts and holomorphic transition functions

• Almost complex structure, complex structure and Newlander-Nirenberg theorem

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

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

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

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

• De Rham cohomology, Dolbeault cohomology, Hodge numbers, Euler characteristic invariant

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

• K¨ahlerforms, K¨ahlerclasses, K¨ahlercone

n • The Fubini-Study metric on CP is a K¨ahlermetric 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)

3 - W. Ballmann, Lectures on K¨ahlermanifolds, Lectures in Math and Physics, EMS (2006). See http://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 and toric geometry, http://arxiv.org/abs/hep-th/0702063

- A. Moroianu, K¨ahlergeometry, 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, Th´eoriede Hodge et g´eom´etriealg´ebriquecomplexe , Cours sp´ecialis´es10, SMF 2002. Translation in english: Hodge Theory and complex algebraic geometry I and II, Cambridge University Press 2002- 3.

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

4 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-Yau manifold (M, J, ω): this is a compact K¨ahler2 manifold such that one of the following conditions is satisfied

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

• c1(M) = 0,

n ∗ • the canonical line bundle KM = Λ T M is trivial, where n = dimC M, • there exists a nowhere vanishing holomorphic n-form.

Note that here the crucial point is to show that any of the last 3 conditions imply 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 of complex Monge-Amp`ereequation and it turns out that one needs to show that if ν is a smooth volume form and [ω] a given K¨ahler class, then there exists 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. One considers the 1-parameter family (0 ≤ t ≤ 1) √ (ω + −1∂∂φ¯ )n = etf+ct ωn √ ¯ R tf+ct n n where Ric(ω) = −1∂∂f and ct is a constant such that M e ω = [ω] . 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 characterize Calabi-Yau manifolds thanks to their holonomy. 2For symplectic or complex non K¨ahlermanifolds, the interested reader will find some engrossing recent achievements in http://arxiv.org/abs/0802.3648 (J. Fine and D. Panov), or http://arxiv.org/abs/math/0205012 (J. Gutowski, S. Ivanov, G. Pa- padopoulos).

5 that S is open (using the implicit function theorem) and closed (this is the hard part, especially the C0 estimate). For the openesse, the ellipticity of the Laplacian of the ambiant metric is crucial, while the closedness is based on Schauder theory and clever a priori estimates. Improvements of this method have been achieved by S. Kolodziej and Z. Blocki for degenerate right hand side using pluripotential theory. On another hand, H.D. Cao proved the Calabi conjecture using K¨ahlerRicci flow.

We give examples of Calabi-Yau manifolds by using a computation of the n first Chern class of a hypersurface of degree d in CP . Unfortunately, we don’t have time to discuss applications to topology (Miyaoka-Yau’s inequality, Fano manifolds are simply connected), to rigidity of the Calabi-Yau theorem (see F. 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 K¨ahlerclasses. Let us fix an ample line bundle L, assume that Lk is very ample3 and ν a smooth4 volume form. We will denote Met(Ξ) the space of smooth hermitian metrics on the vector space or vector bundle Ξ. Then we have

Theorem 1. Let us consider h ∈ Met(L) a smooth hermitian metric over L with positive curvature c1(h) > 0. Then we can define the Bergman function over M dim H0(Lk) X 2 Bk(p) = |Si|hk (p) i=1 0 k R k where Si is an orthonormal basis of H (L ) with respect to M h (., .)ν and p ∈ M. We have an asymptotic when k tends to infinity:

c (h)n c (h)n scal(c (h)) B (p) = kn 1 + kn 1 1 + O(kn−2). k ν ν 2 3In the language of physicists we have a positive polarisation, i.e we are in the context of geometric quantization. 4Actually, one could choose generalize to some extent our results in the non smooth case.

6 The Bergman function is the restriction over the diagonal of the integral kernel of the L2-projection from the space of smooth sections of Lk to the space of holomorphic sections of Lk. The asymptotic formula can be inter- preted as a Riemmann-Roch pointwise formula and the first two terms are simple geometric quantities (scal(ω) stands for the scalar curvature of the metric ω that is the trace of the Ricci curvature Ric(ω)). The proof can be obtained 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 peak sections. Noticing that the Bergman function is depending on the choice of the 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 function Bk,h is constant over the manifold, that is dim H0(Lk) Bk,h(p) = 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,

dim H0(Lk) X dim H0(Lk) |S |2 (p) = i FS(H) V ol (M) i=1 ν

where Si is H-orthonormal basis.

k 0 k k • Hilbν : Met(L ) → Met(H (L )) such that for any h ∈ Met(L ), Z Hilbν(h) = h(., .)ν, M 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-balanced and we get for such a metric Z < Si,Sj >FS(H) ν = δi,j M 5See also T. Bouche’s paper in references.

7 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 of metrics. Donaldson showed that it has a trivial behaviour: Theorem 2. Let ν any given smooth volume form. Then the iterations of Hilbν ◦ FS give a dynamical system with fixed attractive points, unique up to action of U(H0(Lk)). Balanced metrics are limits of the iterations of the Hilbν ◦ FS map. 1 2 1 The proof is very natural. If we set ψz(H) = N log |z|H + N log det H, then we can consider the functional over Met(H0(Lk)), Z Ψν(H) = ψz(H)ν M

Actually Ψν is convex and proper. It is the finite dimensional analog of the so-called Aubin-Yau functional. Its critical points are precisely ν-balanced metrics. Finally, using the arithmetico-geometric inequality and the concav- ity of the log, one can check that an iteration of Hilbν ◦ FS decreases the functional 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-vari´et´escomplexes 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 k tends to infinity. From Theorem 1, we know that if it is convergent, then at infinity we get a metric h∞ such that n (c1(h∞)) = ν i.e we have solved the Calabi conjecture, by prescribing in a given K¨ahler class c1(L) the volume form of the K¨ahlermetric. In order to show the convergence, we can use Yau’s theorem (so this method does not give an alternative proof for the moment). This is quite complicated and use a subtile implicit type function theorem based on the structure of double symplectic quotient (explained later, with the quantification of the Hermitian-Einstein equation). Some references:

8 - Z. Blocki, The complex Monge-Amp`ere equation on compact K¨ahler manifolds, 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-Amp`ere equation and pluripotential theory. Mem. Amer. Math. Soc. 178 (2005)

- J.P. Bourguignon, Premi`eres formes de Chern des vari´et´esK¨ahl´eriennes compactes, Seminaire Bourbaki 507., Lecture notes in mathematics 710 (1977-1978)

- Y.T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and K¨ahler-Einsteinmetrics, Birkhauser, (1987)

- T. Aubin, Nonlinear Analysis on Manifolds, Monge–Amp`ereEquations. 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 Quarterly Jour. 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 m´etriquede Fubini-Study d’un fibr´e lin´eaire positif, Ann. Inst. Fourier (1990). http://www.numdam.org/numdam-bin/fitem?id=AIF_1990__40_1_117_ 0

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

9 3 Numerical approximations of Calabi-Yau met- rics

We have seen in the previous lecture a canonical way to find a ν-balanced metric and that ν-balanced metrics of order k converge when k tends to infinity towards the solution to the Calabi conjecture. Actually, a more precise 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 an integral K¨ahlerclass:

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 to reduce the dimension if possible

0 k 4. Fix a random hermitian matrix H0 ∈ Met(H (L ))

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

The algorithm for finding the ν-balanced metric has an exponential speed of convergence dependant on the smallest positive eigenvalue of the Laplacian of 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 equivalent to k4n, where most of the computations are done to evaluate the Bergman function (essentially a “polynomial” of degree k in n variables) over the whole set of points representing the manifold. In particular it is clear that using this method, one can compute with a single desktop computer any Calabi-Yau metric on complex surfaces (even with no symmetry).

Some examples 1 We discuss the basic example of CP and the Fubini-Study metric. In that case 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).

10 A more interesting case is given by a double cover of the plane branched over the sextic curve x6 +y6 +z6 = 0. We will denote by S this K3 surface. It was studied by S. Donaldson who obtained a formula to compute its volume. dx∧dy 2 Here the nowhere vanishing 3-form is given by w in the chart {w = 2 x6 + z6 + 1}. Over CP , 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}. The 2 double cover S → CP , is the pull-back by f of the covering of the quadric 1 1 ∼ CP × CP over the plane branched along the conic {X2 + Y 2 + Z2 = 0}. Thus, one get a holomorphic isomorphism that sends an open subset of 1 CP × C to an open dense subset of S. By using the elliptic fibration and the symmetries, one obtains explicitly 2 charts that constitue almost an atlas of 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 the manifold with respect to the chosen volume form. The results show a good accuracy of the whole algorithm. It is also very stable (if we modify the volume a little bit, one does not change much the ν-balanced metric, which can be interpreted as a form of stability of the Monge-Amp`ere,see S. Kolodziej). Note that if one linearizes the Hilbν ◦ FS map, one gets around the balanced point that its differential is given by 1 Z X ε 7→ < s , s >< s , s > ε ν dim H0(Lk) α β γ δ γ,δ M γ,δ Using peak sections, one can show that this operator is the quantification of the operator − ∆ω∞ e 4πk where ω∞ is the Ricci flat metric. Hence, as a by product, it is possible to obtain the spectrum of the Laplacian of the Ricci flat metric. This also provides a Newton type method close to the balanced point that gives a refinement of the original method with a better precision.

An even more interesting example to study, is the family of Fermat quin- 4 tics in CP :

5 5 5 5 5 {z0 + z1 + z2 + z3 + z4 − 5ψz0z1z2z3z4 = 0}

11 Then, one can estimate the number of holomorphic sections for O(k). One chooses for the following 3-form using the Poincar´eresidue map,

ψdZ0 ∧ dZ1 ∧ dZ2 Ω = 4 Z3 − 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-Carlo type method. We give some explanations about this idea now. Recall that for a random variable Y on the probability space (Γ, dγ), the expected value R 0 k0 of Y in dγ is E(Y ) = Γ Y dγ. Consider S = {s ∈ H (L ): kskL2 = 1} the unit sphere that we can equip with the Fubini-Study K¨ahlerform. Then, let us choose s ∈ H0(Lk0 ), and define 1 √ T : s 7→ T = −1∂∂¯log |s|2 s 2π which is the current of integration given by Lelong-Poincar´eformula. A lemma of Zelditch and Shiffman (see http://arxiv.org/abs/math/9803052, section 3) asserts that the expected value Z E(T ) = TsdµFS(s) S

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

1 ∗ E(T ) = ι (ω 0 k ∨ ) k k0 F S,PH (L 0 )

0 k0 ∨ where ιk0 is the embedding of the manifold into the projective space PH (L ) . The same applies to current associated with the simultaneously vanishing of p ≥ 1 random section of S. Thus, in order to find points on the manifolds, it is sufficient to intersect the manifolds with lines6 (in the smallest projective space in which one can embed the manifold, i.e the smallest k0 > 1), get the intersection points, and one knows now that they are distributed according to 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-Raphson algorithm that enables to find roots of a polynomial of one variable. Thus

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

12 one obtains the points over the manifold and it is possible to do numerical integration over the quintic. Most of the program presented during my talk can be downloaded from my webpage: http://www.cmi.univ-mrs.fr/~jkeller/Julien-KELLER-progs.html. The precision of the computation of the Ricci flat metrics are in average around 1% in L1 norm (and for a few minutes of computations). The code is written in C++. Feel free to modify it to obtain Ricci flat metrics on other manifolds !

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 K¨ahlermetrics (especially Fano K¨ahler-Einsteintoric 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 for quotients and complete intersections, JHEP 2008. http://arxiv.org/abs/0712.3563

- M. Douglas, R. Karp, S. Lukic, R. Reinbacher, Numerical solution to the 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-Yau manifolds as an exactly soluble superconformal theory.

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

13 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 K¨ahlermanifold7 and E a holomorphic vector bundle on M, one says that the hermitian metric h ∈ 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

n−1 R c1(L) deg(E) c1(E) ∧ µ(E) = µ (E) = = M (n−1)! . L rank(E) rank(E) It is not true that any vector bundle E carries such a metric. On another hand, 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 needs ¯ to 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 be Mumford stable if for any coherent subsheaf F with rank(F) < rank(E), one has 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 (for K¨ahlermanifolds), asserts that if E is an irreducible vector bundle, the exis- tence of a Hermitian-Einstein metric is equivalent to the Mumford stability of E. 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, that it to see ω as the curvature of a hermitian metric on a polarisation L. Much of what we explain in that section can be adapted to the K¨ahlernon-projective case.

14 in topology, or in algebraic geometry through the study of moduli spaces of solutions. For instance, such a correspondance is used in the work of A. Teleman to complete the classification of class VII0 (non-algebraic) complex surfaces with b2 = 1 (Cf. Nakamura and Kato’s conjecture of global spherical shell for the VII class). For general introduction on the Kobayashi-Hitchin correspondance, we recommend to read:

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

- Huybrechts, Daniel and Lehn, Manfred, The geometry of moduli spaces of 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 enables us to construct quasi-projective moduli spaces of vector bundles over a pro- jective manifold (M,L). This is the notion of Gieseker-Maruyama stability that we present now. First of all, we recall the definition of the Hilbert poly- k P i i k nomial for a sheaf F, i.e χF (k) = χ(F ⊗ L ) = i(−1) dim H (M, F ⊗ L ). Remark that by the ampleness of L, this sum reduces to the dimension of the space of holomorphic sections for k large enough.

Definition 3. Let E a holomorphic vector bundle. Then E is said to be Gieseker stable if for any coherent subsheaf F with rank(F) < rank(E), one has χ(F ⊗ Lk) χ(E ⊗ Lk) < , rank(F ) 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. Finally if E ⊗ Lk satifies the previous inequality, we will say that E is k-Gieseker stable (resp. semistable or polystable).

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

15 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 an embedding of the manifold in the Grassmannian Gr(r, N) of r-planes in CN where N = dim H0(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 map µSU(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 group acting on Y by symplectomorphisms (the action preserves the symplectic form). Then, a moment map for the action of G is a map µ : Y → Lie(G)∗ such that: • if we set µX = hµ, Xi (with h, i the dual pairing on Lie(G)) is a Hamil- −→ ∂ exp(tX)(p) tonian function, i.e if X (p) = { ∂t |t=0} is the vector field gen- erated by X, then −→ dhµ, Xi(p) = ιX ωp(., .), where ι is the interior product (contraction of the form with the vector field). • µ is required to be G-equivariant, where G acts on Lie(G)∗ via the coadjoint action. Thus one obtains a homomorphism of Lie algebra.

n Let√ us give an example. If one considers C with the standard form −1 ωstd = dzi ∧ dz¯i, then U(n) acts in an obvious way. For X ∈ Lie(U(n)), 2 √ −→ −1 ∗ one has X (z) = X · z. We check that if we set hµ, Xi(z) = 2 z Xz, then one gets a moment map: √ −1 dhµ, Xi(z)(v) = (v∗Xz + z∗Xv) √2 −1 = (v∗Xz − v∗Xz) 2 = −Im(v∗Xz)

= ωstd(X(z), v).

If we use the coupling (A, B) = −tr(AB) on Lie(U(N)) = {A ∈ M√n(C): ∗ −1 ∗ A + A = 0}, then we can identify the moment map with µ(z) = − 2 zz .

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

∗ ∗ hµ(z), [X1,X2]i = −Im(z X2 X1z) = {hµ, X1i, hµ, X2i}(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)):

Z n  ∗ r  ω (p) Z(p)Z (p) − IdN M N 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 ∈ 0 k 0 k Met(H (E ⊗ L )) such that for (Si) an H-orthonormal basis of H (E ⊗ L ), one has Z ωn(p) < Si,Sj > = δij, M n! where <, > denotes the Fubini-Study metric induced on E ⊗ Lk by H. On another words, it means that the pull-back by ι of the Fubini-Study metric from the Universal vector bundle (over Gr(r, N)) can be identified with the metric on E. Thus, there is a natural definition of balanced metric for vector bundles that correspond to Gieseker polystability. Definition 5. A metric h ∈ Met(E ⊗Lk) is said to be k-balanced if for given 0 k R ωn (Si) an orthonormal basis of H (E ⊗L ) with respect to M h(., .) n! , one has

dim H0(E⊗Lk) X N S ⊗ S∗h = Id . i i rV ol(M) E i=1

0 k A metric H ∈ Met(H (E ⊗ L )) is said to be k-balanced if for (Si) an H- orthonormal basis of H0(E ⊗ Lk), one has ∀i, j, Z ωn(p) < Si,Sj > = δij, M n!

PN N where <, > is the Fubini-Study metric given by i=1 Si < ., Si >= rV ol(M) IdE, with N = dim H0(E ⊗ Lk). Theorem 3. The vector bundle E is Gieseker polystable if and only if for all k >> 0, there exists a k-balanced metric.

17 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 metrics when one assumes the existence of a Hermitian-Einstein one. For that, we remark that balanced metrics appear as zeros of two moment maps and that our problem can we sumed up in finding a point in a double symplectic quotient. Consider the set

∞ k N 1,1 ¯ H = {(s1, ..., sN ,A) ∈ C (M,E ⊗ L ) ⊗ A (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 smooth but we just consider a complex orbit) such that the moment map associated to the Gauge group action is precisely at pH = (s1, ..., sN ,A) ∈ H,

X ∗ µG(pH) = sisi . 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 actually holomorphic sections. Thus, a point in the symplectic quotient H//(G × SU(N)) correspond precisely to a k-balanced metric. In order to find such a 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-Einstein metric. This is a consequence of the asymptotic expansion for the Bergman kernel (generalization of Tian-Bouche’s result).

Proposition 4.1. Assume hE ∈ Met(E), hL ∈ Met(L) with positive cur- 0 k vature. Then if (Si) is an orthonormal basis of H (E ⊗ L ) with respect to R k ωn M hE ⊗ hL(., .) n! and

N ∗ h ⊗hk X E L B(hE, hL) = Si ⊗ Si . i=1 Then when k tends to infinity, √ 1  B(h , h ) = knId + kn−1 −1Λ F + scal(ω)Id + O(kn−2) E L E ω hE 2 E

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

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

 1  B(h ) = Cst × Id + O . k,r E kr

1

−1 Then the metric hk,r × IdE + O kr gives the expected point in H//G, and we call it almost-balanced. Finally, in order to obtain a point in (H//G)//SU(N), we just need in an SL(N)-orbit. Thanks to the next result due to Donaldson, we know that if the starting point is not too far from 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 group acting on Y by symplectomorphisms. Assume that µ is a moment map for the G action on (Y, χ). Let σy : Lie(G) → TYy be the infinitesimal action. ∗ Assume that Qy = σyσy is invertible. Consider a point y0 ∈ Y such that

δ •| µ(y0)| < λ √ −1 −1θ • ||Qy || < λ for all y = e y0 avec |θ| < δ. √ −1θ1 Then, there exists y1 with y1 = e y0, |θ1| < λ|µ(y0)| and µ(y1) = 0.

−1 With Proposition 4.2, we can control the operator norm ||Qy || at the 2 almost-balanced metric hk,r. This use some classical L estimates. We don’t present details about it since it is conjectured that one can still improve technically this part. Finally, with Proposition 4.3, we have all the ingredients to obtain

Theorem 4. Assume that E has a Hermitian-Einstein metric. Then there k −k exists a sequence of balanced metrics hk ∈ Met(E ⊗ L ) such that hk ⊗ hL converges 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

8 1 Actually h∞ is just almost-Hermitian Einstein, that is ΛFh∞ + 2 scal(ω)Id = cst × Id

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

Some references :

- Y.T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and K¨ahler-Einsteinmetrics, 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 Bergman kernels, Birkhauser, 2007.

- Mumford, D. and Fogarty, J. and Kirwan, F., Geometric invariant theory, 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 projective manifold, 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 work with H. D. Cao.

We have seen that if we set (M,L) a polarized manifold, ν a smooth volume form, there exists for k >> 0 a notion of ν-balanced metric. They k are fixed points of the operator FS ◦ Hilbν acting on Met(L ). Another way of presenting the notion of ν-balanced metric is to introduce a moment map N µ : CP k → iu(N + 1) for the U(N + 1) action. Given homogeneous unitary coordinates, one sets µ = (µ)α,β where z z (µ) [z , ..., z ] = α β . α,β 0 N P 2 i |zi| Then, given an holomorphic embedding ι : M,→ PH0(Lk)∨, we can consider the integral of µ over M with respect to the volume form: Z µν(ι) = µ(p)ν(p) M which provides a moment map for the U(N + 1) action over the space of all bases of H0(Lk). Actually, there is a K¨ahlerstructure on that space isomorphic to GL(N + 1), and U(N + 1) acts isometrically with the moment map which is essentially √  tr(µ (ι))  − −1 µ − ν Id . ν N + 1 N+1 The embedding ι is ν-balanced if and only if tr(µ (ι)) µ0(ι) := µ (ι) − ν Id = 0. ν ν N + 1 N+1 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). 0 On another hand, seen as a hermitian matrix, µν(ι) induces a vector field N on CP k . Thus, inspired from the work of J. Fine, we study the following flow dι(t) = −µ0(ι(t)) (1) dt ν 21 and we call this flow the the ν-balancing flow. To fix the starting point of this flow, we choose a K¨ahlermetric ω = ω(0) and we construct a sequence of hermitian metrics hk(0) such that ωk(0) := c1(hk(0)) converges smoothly to ω(0) providing a sequence of embeddings ιk(0) for k >> 0. Such a sequence of embeddings is known to exist by a result of Tian. For technical reasons, we decide to rescale this flow by considering the following ODE.

dι (t) k = −kµ0(ι (t)) (2) dt ν k that we call the rescaled ν-balancing flow. Of course, we are interested in 1 ∗ the behavior of the sequence of K¨ahler metrics ωk(t) = k ιk(t) (ωFS) when t and k tends to infinity.

∞ Theorem 5.√For any t, the sequence ωk(t) converges in C sense to the ¯ solution ω + −1∂∂φt of the Monge-Amp`ere equation ∂φ ν t = 1 − √ (3) ¯ n ∂t (ω + −1∂∂φt) with φ0 = 0 and ω = limk→∞ ωk(0). We call this flow the ν-K¨ahlerflow.

5.1 Study of the limit of the balancing flow

In that section, we assume that the sequence ωk(t) is convergent and we want 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 embedding 0 k ∨ ∗ ι : M,→ PH (L ) . The induced infinitesimal change in ι ωFS is pointwisely given by the potential tr(Hµ). Thus, the corresponding potential in the case of the ν-balancing flow is 0 β = −tr(µνµ). Since we are rescaling the flow in (2) and considering forms in the class 2πc1(L), we are lead to understand the asymptotic behavior when k → ∞ of 0 the potentials βk = −ktr(µνµ). Using Tian-Bouche asymptotic expansion, we can prove:

k Proposition 5.2. Let hk ∈ Met(L ) be a sequence of metrics such that ωk := 1 k c1(hk) is convergent in smooth topology to the K¨ahlerform ω. Then the

22 potentials βk induced by the metrics Hilbk(hk) converge in smooth topology to the potential ν 1 − , ωn that is the potential of Equation (3).

Let us give a sketch of the proof. Let (si) be an orthonormal basis of 0 k H (L ) with respect to the metric Hk := Hilbk(hk). The balancing potential at p ∈ M is Z ! X hsα, sβi(q) δαβ hsα, sβi(p) n βk(Hk) = − N − N ν (q), P k 2 Nk + 1 P k 2 M α,β i=1 |si(q)| i=1 |si(p)| where h., .i stands for the fibrewise metric hk. By Riemann-Roch theorem, n n−1 Nk = k V ol(L) + O(k ). From Tian’s theorem and the fact that ωk is convergent, we obtain

n Z n ! Nk k hsα, sβi(q)hsα, sβi(p) k ν n βk = − ωk (q) N PNk 2 kn PNk 2 ωn(q) k i=1 |si(p)| M i=1 |si(q)| k    Z     1 hsα, sβi(q)hsα, sβi(p) 1 ν n = 1 − 1 + O n 1 + O n ωk (q) k M k k ωk (q) But now, from Fine’s paper (Theorem 18), one knows the asymptotic behav- ior of the quantification operator 1 Z X Q (f)(p) = hs , s i(q)hs , s i(p)f(q)ωn(q). k kn α β α β k M α,β

C m m Precisely, kQk(f) − fkC ≤ k kfkC for an independent constant C > 0. Then, for k → ∞, ν  1 βk(Hk)(p) = 1 − n Qk 1 + O ωk k 1

One has convergence of Qk 1 + O k 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 by Equation (2) converges in smooth topology to a metric ωt and, moreover, that 1 this convergence is C in t. Then the limit ωt is a solution to the flow (3) starting at ω0 = limk→∞ ωk(0).

23 5.3 Study of the ν-K¨ahlerflow We are now interested in the flow ∂φ ν t = 1 − √ (4) ¯ n ∂t (ω + −1∂∂φt) over a compact K¨ahlermanifold (not necessarily in an integral K¨ahlerclass), where φ0 = 0 and ω is a K¨ahlerform in the class [α]. Of course, this can be rewritten as √ ¯ n 1 f n (ω + −1∂∂φt) = e ω (5) ∂φt 1 − ∂t where f is a smooth (bounded) function defined by f = log(ν/ωn). We are 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 of cohomology class, we obtain directly √ ∂[(ω + −1∂∂φ¯ )] t = 0, ∂t √ ¯ which shows√ that the K¨ahlerform 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 Calabi conjecture. For instance, by maximum principle, one gets:

∂φt 1 0 Proposition 5.4. The function and ∂t and ∂φt remain bounded in C 1− ∂t norm along the flow given by Equation (5).

We denote ∆ the Laplacian with respect to the K¨ahlerform ω and by  ν ∂ the operator n ∆t − . Applying the ideas of Yau and Cao (i.e the maximum ωt ∂t principle to ∆φ using the operator , one gets:

Lemma 5.1. One has a lower and upper bound for ∆φt. Then, there is an a 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 to the J-flow studied by S.K. Donaldson, X.X. Chen, and B. Weinkove.

Some references :

24 - H-D. Cao, Deformation of K¨ahlermetrics to K¨ahler-Einsteinmetrics on compact K¨ahlermanifolds, Invent. Math, 81 (1985).

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

- X.X. Chen and S. Sun, Space of K¨ahler metrics (V)— K¨ahlerquanti- 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 Bergman kernels, 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 the complex Monge-Ampere equation I. Comm. pure appl. math, (1978).

25 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- Koll´artheorem, and properness of the K-energy. Applications to the toric case.

• Applications. Stability for projective manifolds and Yau-Tian-Donaldson conjecture.

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