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About the Calabi-Yau Theorem and Its Applications 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; !): this is a compact K¨ahler2 manifold such that one of the following conditions is satisfied • Ric(!) = 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 [!] a given K¨ahler class, then there exists a unique smooth potential φ solution of the highly non linear PDE: p (! + −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) p (! + −1@@φ¯ )n = etf+ct !n p ¯ R tf+ct n n where Ric(!) = −1@@f and ct is a constant such that M e ! = [!] . 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 2 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) = jSijhk (p) i=1 0 k R k where Si is an orthonormal basis of H (L ) with respect to M h (:; :)ν and p 2 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(!) stands for the scalar curvature of the metric ! that is the trace of the Ricci curvature Ric(!)). 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 2 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 2 M. Then, we can introduce two natural maps: • FS : Met(H0(Lk)) ! Met(Lk) such that for all p 2 M, dim H0(Lk) X dim H0(Lk) jS j2 (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 2 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 z(H) = N log jzjH + N log det H, then we can consider the functional over Met(H0(Lk)), Z Ψν(H) = z(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).
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