Perspectives and Advances in Kähler Geometry

Transcription

1 Perspectives and Advances in Kähler Geometry Shing-Tung Yau Harvard University 30th Shanks Lecture, Vanderbilt University May 19, 2015

2 1. Introduction of the concept of Kähler manifolds The foundation of the theory of complex manifolds went back to Riemann who wanted to use geometry to understand complex analysis. But it was Erich Kähler in Hamburg who defined the fundamental concept of Kähler geometry in S.-S. Chern arrived at Hamburg university in September of 1934, where he was the only audience of the lectures given by Kähler on the Cartan-Kähler theory of exterior differential system. In 1978, I had the privilege to be introduced by Chern to meet Kähler for a cup of coffee during the 1978 ICM in Helsinki. 2 / 70

3 One should mention that the Hermitian form of a Hermitian metric was already introduced by E. Cartan in Lecons sur les invariants intégraux, Hermann, Paris (1922). By looking at the example of the Poincaré metric, Kähler demands the Hermitian form to be closed. And he derived that locally, such a form must be of some potential. And such Hermitian metric is called Kähler metric. The elegant theory based on Kähler metric has spectacular influence in geometry and physics all the way up to now. 3 / 70

4 Many people, including myself, did not realize that Kähler did many remarkable calculations in his very first paper on Kähler geometry. He found that the Ricci form can be written as 2 R k l = (log det g z k z ). i j l He knew that the Ricci form gave a globally defined closed form on the manifold. The wedge product of copies of the Ricci form and copies of the Kähler form can be integrated to give numbers similar to invariants such as Euler number. He considered this an analogue of the Gauss-Bonnet theorem. 4 / 70

5 A most remarkable fact was that Kähler has already observed the importance of Kähler-Einstein metric, which indicates the tremendous interest of geometers on general relativity. He wrote down the equation of the metric which says that in the right coordinate, the volume form is equal to the e ck, where K is the Kähler potential satisfying g i j = 2 K z i z j and c is the scalar curvature. This was discussed way before Chern s famous paper on Chern classes. Apparently Kähler did not know the idea of canonical class studied by Italian algebraic geometers. (In 1931, de Rham has proved the de Rham theorem which implies that the closed form defined by the Ricci form represents a cohomology class.) 5 / 70

6 In fact, the globally defined Ricci form gives the first Chern class of the manifold which was the canonical class studied by projective geometers. If he knew this definition, it would have been natural for him to discover the conjectures of Calabi which was made much later in In any case, he pointed out the equation that needs to be solved for Kähler-Einstein equation should be considered as higher dimensional generalization of the works of G. Giraud (1918, in Comptes Rendus) and A. Bloch (1927, in L Enseignement). 6 / 70

7 2. Hodge theory Hodge was interested in the Kähler metric of projective algebraic manifolds induced by the Fubini-Study metric, without knowing the works of Kähler. Hodge decomposed the forms into (p, q)-forms. The most important observation was that the projection operator that projects k forms to (p, q)-forms with p + q = k commutes with the Lapacian operator acting on forms. This means that the kernel and the eigenforms of the Laplacian can be decomposed into sums of (p, q)-forms. Harmonic k forms are sums of harmonic (p, q)-forms with p + q = k. This remarkable fact, in conjunction with various Kähler identities gives very strong constraints on the cohomology of Kähler manifolds. 7 / 70

8 Immediately after the papers of Hodge, K. Kodaira in Japan published several papers (e.g., Harmonic fields in Riemannian manifolds, Ann. of Math. 1949) where he also considered manifolds with boundaries. He gave correct proof of the existence of the harmonic forms and the whole theory is sometimes called the Hodge-Kodaira theory. The proof of Hodge decomposition becomes much more transparent when Milgram-Rosenbloom published Harmonic forms and heat conduction in Proc. Nat. Acad. Sci. USA (1951), the heat equation method exerted deep influence in the future of analytic methods in geometry, including the proof of local index formula and the works of Eells-Sampson and Hamilton. 8 / 70

9 A fundamental question in Kähler geometry is to understand the splitting of the k-th cohomology into cohomologies of type (p, q) with p + q = k. In principle, the position of type (p, q) cohomology should determine the complex structure of the manifold. This type of question can be considered a generalization of the Torelli theorem which Torelli proved in Rend. Acc. Lincei (1913). It can be regarded as the one-dimensional version of this question. There is a long history of the Torelli type theorems in the past fifty years. 9 / 70

10 There are geometric proofs of Torelli theorem for curves and K3 surfaces. The most notable one was the due to Andrei Todorov and Siu. When I finished the proof of Calabi conjecture, I proposed to Todorov to use the canonical Ricci-flat metric to prove the injectivity and surjectivity of the period map of K3 surfaces. It is quite likely that the CY metric can be used to prove suitable version of Torelli in higher dimensional CY manifolds. 10 / 70

11 Note that the calculation of the position of the vector subspace of closed (p, q)-forms within the topological vector space of closed k-forms (k = p + q) depends on understanding the vector given by the integrals of the closed (p, q)-forms over a fixed base of topological cycles. The computation of such periods for Riemann surfaces went back to Euler, Gauss, Riemann, Jacobi, Abel and others. Higher dimensional version started by the turn of 18th century to 19th century. 11 / 70

12 When the moduli space has dimension greater than one, one needs to generalize the Picard-Fuchs equation to a system of equations. Gelfand et al. initiated in the 90 s the study of multivariable differential systems with regular singularities in an attempt to generalize Euler integrals. But the GKZ theory turns out to be not enough to treat the problem of Picard-Fuchs equations. Recently Bong Lian, An Huang and I worked with others to introduce the concept of tautological system, which generalizes GKZ, in order to solve this problem. From the recent development of mirror symmetry, it is most likely that explicit form of Calabi Yau metric, to be discussed later, could be written in terms of periods of integrals. 12 / 70

13 Hodge observed that the cohomology class associated to any algebraic cycle must be of (p, p) type and he asked the converse in 1950 during his address in the ICM held in Harvard. The only positive partial answer for the Hodge conjecture was the paper of Chern in 1954 that every Chern classes of holomorphic bundles are algebraic. The Hodge conjecture is perhaps the most important problem in projective geometry that is still not solved. 13 / 70

14 In the 1940s, Bochner observed that positivity of Ricci curvature implies the vanishing of the first Betti number, based on computation of the Laplacian of the norm of the harmonic one-form. Immediately afterwards, Kodaira generalized the vanishing theorem of Bochner to Kähler manifolds. The vanishing theorem of Kodaira is the most powerful tool in Kähler geometry. In particular he found the criterion to determine which Kähler manifold is projective algebraic. 14 / 70

15 3. Chern classes and Riemann-Roch formula When Hirzebruch proved the higher dimensional Riemann-Roch formula, the Chern class is an essential tool. The Hirzebruch-Riemann-Roch formula and the Chern class have become the most fundamental part of global geometry up to modern days. The expression of Chern classes by curvature of the bundle is tremendously important. However the major effort had been devoted to the understanding of the first and second Chern classes. It is still difficult to understand the differential geometric meaning of higher Chern forms. 15 / 70

16 The Hirzebruch-Riemann-Roch formula and its extension to the Atiyah-singer index formula is the most powerful identity in mathematics. Hirzebruch discovered the concept of Chern character which gives a ring homomorphism from vector bundles to cohomology. Then he defined multiplicative genus, which became fundamental invariants for later development. His observation that the  genus is integral is particularly important for the development of index theory. According to I.M. Singer, he was very curious about this statement after he was told by Atiyah about it. It took him two weeks to find a right definition of the Dirac operator whose index gives rise to the  genus. With that, he was confident about the statement of the index formula. 16 / 70

17 The proof of Riemann-Roch formula by Hirzebruch works only for algebraic manifolds. But an important corollary of Atiyah-Singer index formula is that it also works for arbitrary complex manifolds. This fact is fundamental for the classification of complex surface due to Kodaira. According to the classification of Kodaira, complex surfaces with even first Betti number can all be deformed to algebraic surfaces. Only a few complex surfaces have odd first Betti number. Hence we may say that most complex surfaces are Kähler. 17 / 70

18 However, the situation is quite different for dimension greater than two, Hirzebruch-Riemann-Roch does not seem to produce enough constraints for an almost complex manifold to admit integrable complex structure. I made the conjecture that for dimension greater than two, any almost complex manifolds would admit an integrable complex structure. It may be overoptimistic, but no counterexample has been found. 18 / 70

19 4. Kähler geometry The fact that the Hodge Laplacian commutes with the projection on (p, q)-forms provides a very powerful tool. The Hodge structure defined on de Rham forms descends to cohomology level which gives a vast generalization of the Torelli theorem. Projection operator can be considered as zeroth order differential operator. It will be interesting to find nontrivial differential operators that commute with the Hodge Laplacian. If such operator exists, it will split the eigenfunctions of the Hodge Laplacian in a different way and hence give interesting structural properties of such manifolds. In particular, the graded ring of such differential operators will then act on harmonic forms or eigenforms. 19 / 70

20 The Kähler structure gives rise to a set of Kähler identities which can be used to prove the -Lemma, which is supposed to be equivalent to the degeneracy of the E 2 term of the Hodge-Frölicher spectral sequence and the existence of a Hodge structure on the cohomology groups. -Lemma played an important role in Kähler geometry. For example, it was used by Deligne, Griffiths, Morgan and Sullivan to prove that the minimal model of the rational homotopic type of a Kähler manifold is determined by algebraic structures of its de Rham complex. 20 / 70

21 I propose to prove that every complex manifold that satisfies -Lemma can be deformed to a Kähler variety, possibly with singularity. Hopefully this may give some hint of what kind of complex manifolds may support a Kähler structure. Dennis Sullivan had introduced an interesting idea based on Hahn-Banach theorem to construct geometric structures. I recommended this to Siu to prove every K3 surface is Kähler. Harvey and Lawson also used similar idea to prove the following characterization of Kähler manifolds: If a compact complex manifold carries no positive (1, 1)-currents which are (1, 1)-components of boundaries, then this manifold supports a Kähler metric. 21 / 70

22 Note that the method of Kodaira is to choose a Hermitian metric on the line bundle to have the right curvature to prove vanishing theorem. This was generalized by Andreotti-Vessentini and Hörmander to noncompact complex manifolds, using the concept of weight functions, which may be considered as conformal change of Hermitian metrics on the bundles. 22 / 70

23 While Kodaira used the idea of blowing up to create holomorphic sections of line bundles, one can use singular weight function to arrive at the same effect. This was used by Siu-Yau to create holomorphic functions of certain growth on simply connected complete Kähler manifolds with nonpositive curvature. In fact, we produced enough of them to give a biholomorphic map from the manifold to complex Euclidean space, under certain curvature decay condition. By choosing the weight functions to be singular in different way, one can produce holomorphic sections with various properties. The concept of multiplier ideal sheaf was later introduced by Alan Nadel to handle more complicated analytic problems. 23 / 70

24 The idea of Bochner-Kodaira is based on the Hilbert space of square integrable functions. The subspace of square integrable holomorphic functions are of course very interesting. Stefan Bergman (1922) studied this space extensively by considering an orthonormal basis of this Hilbert space. He found the remarkable reproducing kernel function bearing his name in this space and found that the of the Kernel function is an intrinsically defined Kähler metric depending only on the complex structure of the manifold. Bergman studied the metric extensively on domains in complex Euclidean space with an intention to generalize the Riemann mapping theorem. 24 / 70

25 Shoshichi Kobayashi gave an elegant formulation of the Bergman metric in terms of embedding into complex projective space. I was rather interested in this formulation and was trying to understand how to approximate Kähler metrics by the Fubini-Study metrics. Naturally, we need to assume the Kähler class to be rational in order to find an embedding into complex projective space. Besides an attempt to generalize the previous theorem with Siu, I was also interested in understanding the relation between stability of manifolds and existence of Kähler-Einstein metrics. 25 / 70

26 Since the basic definition of stability of a manifold in the geometric invariant theory is obtained by looking at the projective embeddings of the manifold, I want to divide the problem into two steps. The second step would be to find good stability to relate to the existence of Kähler-Einstein metric. But the first step is to prove that any Kähler metric, whose Kähler class is rational in the second cohomology class, can be approximated by a suitable Bergman metric, which depends on the choice of the line bundle and the Hermitian inner product on the sections of the line bundle. The idea is pretty clear from the argument of Siu and I on the embedding of complete Kähler manifolds with nonpositive curvature. (I later gave this problem and the idea of solving it as a thesis topic to Tian.) 26 / 70

27 Just like the Kodaira embedding theorem, one needs a high tensor powers of a positive line bundle to give the embeddings of the manifolds to achieve the approximation of the metric. In the case of Kodaira embedding, there was a great improvement due to Matsusaka, who proved the very ampleness of ml for an ample line bundle L on an n-dimensional projective variety X when m is no less than a bound depending only on the intersection numbers L n and K X L n 1. The same question is very important for a compact Kähler manifold with zero Ricci curvature, i.e., Calabi-Yau manifolds. 27 / 70

28 If we want to compute the Ricci-flat metrics on a Calabi-Yau manifold, we need to know the power of the line bundle that is necessary for the normalized induced Fubini-Study metric to approximate the given Ricci-flat Kähler metric. When the power is getting large, the target complex projective space will have bigger and bigger dimension. That means the computation of the Ricci-flat Kähler metric will be costly. Most important of all, we need to know the exact error we have made in the computation of the Calabi-Yau metrics. 28 / 70

29 From here, we see that Kähler metrics with rational Kähler classes are very close to algebraic geometry. It will be very important to understand those Kähler metrics whose Kähler classes are not rational. There are two types of irrational Kähler classes: those that can be approximated by rational Kähler classes and those that cannot be approximated. For the first type, there are various methods to see how the nearby rational Kähler metrics converge to it. The second class of Kähler metrics is very interesting and not much is known about them. How do we build a model for such metrics? 29 / 70

30 An important question related to this consideration is the characterization of the Kähler cone. While there were important works due to D ly-Paun on the numerical characterization of Kähler cone, we like to know much more detailed structure of the singular Kähler metric representing the boundary class. Kähler cone for CY manifold is rather special as Wilson proved that the cone structure is invariant under deformation, modulo some simple operations. Tosatti-Weinkove and Tosatti-Collins have done beautiful works in representing boundary points of Kähler cone by singular CY metrics. The study of singular Calabi-Yau metric and the noncompact version was initiated by me in 1976, at the time when I solved the Calabi conjecture. 30 / 70

31 When the Kähler class is rational, we can clear up its denominator and define a holomorphic line bundle whose first Chern class is an integral multiple of the Kähler class. If the Kähler class is not rational, we can still find an analytic object of the following type: Given an open cover {U i } of the complex manifold, on U i U j, there is a holomorphic function f ij which does not define a Cěch cycle unless the Kähler class is integral. However, their absolute values f ij does define a Cěch cycle. 31 / 70

32 Hence, we have a real line bundle where the transition function is defined by f ij. For this line bundle, it makes sense to talk about sections f so that log f is pluriharmonic. If we have enough such sections f i, so that log f i is positive definite, the resulting metric is in the Kähler class of the original metric. It is interesting to know under what conditions can we produce enough sections f i with log f i pluriharmonic, such that the map (f 1, f 2,..., f n ) defines a map from the complex manifold into the quotient of (R + ) n under the diagonal action of positive scalars. (Hence it is a simplex.) If we have enough sections of this type, we can replace most discussions in projective geometry by such sections. 32 / 70

33 Another approach towards non-algebraic Kähler manifolds is to consider holomorphic line bundles L with c 1 (L)/n very close to the Kähler class for large n. Following Donaldson s work on symplectic geometry, one may consider almost holomorphic sections of such line bundle L n and it should be possible to use such sections to define concepts of stability of the Kähler classes. Note that the zero sets of the sections are not necessary holomorphic. If one can make good control of the convergence of such cycles when n, one may obtain a holomorphic cycle with infinite volume, similar to the holomorphic cycle with irrational slope in a complex torus. Finding a universal embedding space for a Kähler manifold is a very interesting question. Perhaps we can try to look into limit of the maps defined above, putting into the balanced condition in the complex projective space so that the Kähler manifold can be embedded into an infinite dimensional projective space. 33 / 70

34 5. Kähler-Einstein metric and stability In 1976, Aubin and I proved independently that for compact Kähler manifold with negative first Chern class, Kähler-Einstein metric exist. And this was extended by S.-Y. Cheng and me to much more general class of manifolds, including complete noncompact Kähler manifolds with infinite or finite volume. The manifold can allow singularities. The theorems have many applications to algebraic geometry and were followed by many people. I also proved the original full conjecture of Calabi which implies the existence of Ricci-flat Kähler metrics on manifolds with zero first Chern class. 34 / 70

35 Noncompact version was described by me in my talk in 1978 ICM in Helsinki. Some of the works were written up with Tian later. The most general noncompact case has not been settled. In the above cases, we do not need to assume any stability of the manifold for existence of Kähler-Einstein metric. Hence it is a corollary that all these manifolds are K-stable in the sense to be described later. 35 / 70

36 It was proposed by me in early 1980s that Kähler-Einstein metric should exist on a Fano manifold if and only if the manifold is stable with respect to the polarization in a suitable sense. There is no doubt that such conjecture should cover the case of Kähler metric with constant scalar curvature. In the case when the scalar curvature of a Kähler-Einstein metric is not zero, the Kähler class is either plus or minus of the first Chern class, hence integral. In the Calabi-Yau case, the elements in H 2,0 can be represented by parallel forms and hence we can rotate the complex structure to make sure the Kähler class can be approximated by integral classes. 36 / 70

37 For general Kähler manifold, the Kähler class may be transcendental and we expect that as long as the manifold is stable in suitable sense, the Kähler class may admit csck metrics. However, definition of stability always involved some holomorphic embeddings into complex projective spaces. That was the reason that I made the conjecture that every Kähler metric with rational Kähler class can be approximated by induced Fubini-Study metrics. This was the thesis topic I gave to Tian and I advised him to use the peak function method that Siu and I developed. In that case, I also noticed a special position of the embedding is needed as the group of projective transformation acts on the embeddings. 37 / 70

38 I was motivated by my work with Peter Li on estimates of the first eigenvalue of the Laplacian. In our joint paper with Bourguignon, we introduced the concept of balanced condition in terms of the Lie algebra of the projective group. (We put the center of gravity of the manifold to be at the origin.) I suggested to my former student Huazhang Luo to use this balanced condition to study the stability of projective manifolds. He modified our balanced condition by choosing the measure to be the induced measure from Fibini-Study metrics. Actually when Shouwu Zhang was writing his thesis in Columbia, he wrote to me and I told him about my conjecture; he did excellent work by using the concept of height in Arakelov geometry. 38 / 70

39 In any case, most of these ideas were discussed in my students seminars where Tian and others attended. The ideas clarified many problems towards the solution of my conjecture. Some simple ideas were introduced to prove the existence of Kähler-Einetein metrics on Fano manifolds. They were based on Hörmander s integral estimates which we called α-invariants. It could be effective for simple Fano manifolds. But the ultimate achievement was made by Chen-Donaldson-Song who settled my conjecture on the existence of Kähler-Einstein metrics on Fano manifolds. It started with the existence of Kähler-Einstein metrics that have simple singularities along divisors. These were initiated in my first paper on the proof of Calabi conjecture. 39 / 70

40 On the other hand, little is known about the corresponding conjecture for the existence of csck metrics, besides the case of complex toric surfaces due to Donaldson. The problem here is that we need to know how to define the concept of stability for general Kähler manifolds, which cannot be approximated by rational Kähler classes. For the problem of csck metrics in rational classes on projective varieties, Székelyhidi introduced an extended notion of K-stability based on filtrations of the coordinate ring, and proved that it is necessary. 40 / 70

41 Once we have a canonical Kähler metric in a given fixed Kähler class, there is much we need to learn about the relationship between complex geometry and the metric geometry. The first major application of Calabi conjecture was to apply the metric to study the Chern classes through the curvature. A very important fact is that for Kähler-Einstein metrics, the first Chern class c 1 is a constant multiple of the Kähler-class ω, while c 2 ω n 2 give rise to the L 2 -norm of the curvature tensor of the Kähler metrics. Hence we obtained inequalities between Chern numbers and proved that equalities gave rise to space forms. An important consequence is that they give information about the fundamental group which is difficult to obtain through methods of algebraic geometry. 41 / 70

42 The canonical Kähler-Einstein metrics give a way to define the canonical Weil-Petersson metric on the moduli space of Kähler manifolds, which enjoys certain curvature properties. Such facts are very important in the study of mirror symmetry in string theory. A very interesting question raised in mirror symmetry is that it is natural to complexify the Kähler class whose imaginary part is call B-field. But we have no good Riemannian geometric interpretation of such a B-field at this moment. I believe this is a worthwhile direction to explore. 42 / 70

43 One needs to construct invariants to understand various kind of stability related to existence of Kähler-Einstein metric suggested by my conjecture: given a Kähler metric w(0), we can form a new metric w(1) by w + i u, where u is any real valued function as long as w(0) + i u is positive. Each Kähler metric w(0) and w(1) created a Ricci form Ric(0) and Ric(1) respectively. If we are dealing with Kähler-Einstein metric with Kähler class being ±c 1, all these forms are in the same classes (after normalizing their signs), hence their difference can be written as of some functions. 43 / 70

44 What happens is that we have a set of (1, 1)-forms and a set of potentials. When there is a vector field on the manifold, we can differentiate those potentials by this vector field and create new set of functions. The forms and the potentials generate a ring and we can integrate those that are of top dimension and find invariants of the manifold. Some of these invariants appeared in my proof of the Calabi conjecture. Some of them were used by Futaki to form the Futaki invariant and by Mabuchi to form the Mabuchi energy. Invariants are useful only if they give information about the manifold when deformed. This is useful for the definition of stability, estimate in continuity argument and Hamilton s Ricci flow. A mirror version of these invariants are interesting. It would involve holomorphic 3-form and its deformations. 44 / 70

45 Probably the most important invariant in Kähler-Einstein geometry is the Futaki invariant of projective manifolds. (This is an invariant attached to Lie algebra of the automorphism group. There were works in this direction due to Calabi, Lichnerowicz and Matsushima.) One can deform a projective manifold and study the behavior of its Futaki invariant. This was used effectively by Donaldson to define K-stability. Phong, Sturm and their students have made substantial contribution to the subject. We refer the readers to their paper Lectures on stability and constant Scalar curvature which is a good survey. 45 / 70

46 The algebro-geometric concept of K-stability introduced by Donaldson is still mysterious and difficult to calculate. Kähler-Einstein metric is, at this moment, the best way to understand them. We hope that methods of algebraic geometry can provide a different angle. 46 / 70

47 Calabi (Seminar on differential geometry, S.-T. Yau ed. 1982) found that if one minimizes the L 2 -norm of the scalar curvature for Kähler metrics in a fixed Kähler class, the resulting extremal metric satisfies a very nice equation: The holomorphic part of the gradient of the scalar curvature is a holomorphic vector field. Hence if the manifold does not support holomorphic vector field, the scalar curvature is constant. And it is a simple observation that if the scalar curvature of the Kähler metric is constant, the first Chern form of the metric is harmonic. It is believed that K-stability, based on the polarization given by this Kähler class, implies the existence of extremal Kaḧler metric. 47 / 70

48 Uniqueness of Kähler-Einstein metric in the Fano case was initiated by Bando Mabuchi. And this was extended by Donaldson, Chen and others to more general cases of constant scalar curvature. However, the existence is still a major problem in Kähler geometry. It should be true that in a Zariski open set of the complexified Kähler moduli, the real Kähler class admits metrics with constant scalar curvature. At this moment, there is no good speculation of what a complexified Kähler metric should be. Perhaps one should study what is a good concept of complex symplectic manifold first. 48 / 70

49 A very interesting question about extremal metric is how they behave when the manifold is an algebraic manifold of general type. There is a canonical singular Kähler-Einstein metric on algebraic manifolds of general type, constructed by the author and Tsuji independently. One would be interested to study its relationship to csck metric. It is also interesting to understand extremal Kähler metric when the manifold is a fibered space over a manifold of general type, with fibers given by rational manifolds or Calabi-Yau manifolds. 49 / 70

50 Given any canonical metric, we can create invariants from the metric or the natural operators defined by the metric. The canonical metrics are very useful to understand the moduli space of polarized algebraic manifolds and their compactifications. This is being carried out by many scholars. The convergence depends on weak convergence of Gromov-Hausdorff. But it is best to keep it within the category of complex varieties. An interesting application of compactness for Kähler-Einstein metrics is to prove that there are only finite number of components of moduli space of Kähler-Einstein manifolds with given diffeomorphic type. In particular, one likes to know whether there are only finite number of Chern numbers that may appear on algebraic manifolds with ample canonical line bundle and with given diffeomorphic type. (Of course, one can ask similar questions for CY and Fano manifolds.) 50 / 70

51 A very interesting question is to compare the spectra of their Laplacians acting on functions or forms. Among the class of Kähler-Einstein metrics, the spectrum should determine the metric and the complex structure. Ray and Singer constructed their invariants by looking at at a combination of the derivatives at zero of the zeta functions of the Laplacian acting on differential forms of suitable degrees. It is an interesting question to find out whether the Ray singer invariant can be defined solely by the complex structure without using zeta functions. In other words, it would be nice to find a complex analogue of the Ray-Singer conjecture on the equality between Reidemeister torsion and Ray-Singer invariant. We need to find a complex analogue of the Reidemeister torsion. 51 / 70

52 Certain combinatorial invariants coming from covering of the manifold by affine open sets may be considered. the real version of Ray singer invariant is related to Chern-Simons invariants. Would the complex version of Ray-Singer invariants be related to the complex version of Chern-Simons invariants? In the theory of BCOV (Bershadsky, Cecotti, Ooguri, and Vafa), the quantum Kodaira-Spencer theory, Ray-Singer invariant appears in mirror version of the higher genus enumerative geometry. It is certainly an important invariant to be understood. 52 / 70

53 There is another type of curvature called holomorphic sectional curvature. For a Kähler manifold with negative first Chern class, a degenerate Kähler metric with negative holomorphic sectional curvature may exist outside the subvariety which is a union of rational curves and elliptic curves. Damin Wu and I proved some form of the converse: if the compact projective manifold admits a Kähler metric with negative holomorphic sectional curvature, then it admits a metric with negative Ricci curvature. This problem makes sense for Finsler metric and was studied by Steven Lu and I using the argument of Miyaoka for algebraic surfaces with c 2 1 > 2c / 70

54 For rational surfaces, Hitchin demonstrated the existence of Kähler metric with positive holomorphic sectional curvature. It is quite possible that similar statement is true for higher dimensional rational manifolds too. On the other hand, a CY manifold cannot support a metric with positive holomorphic sectional curvature or a metric with negative holomorphic sectional curvature. 54 / 70

55 A very important idea in complex surface in the last twenty years is the Seiberg-Witten invariants which can be used to prove that algebraic surfaces of general type cannot be diffeomorphic to rational surfaces. Claude LeBrun found many nice applications of Seiberg-Witten invariants to study topological and geometrical properties of compact 4-dimensional manifolds, and the existence of Einstein metrics. 55 / 70

56 In complex two dimension, both Chern numbers and Seiberg-Witten invariants depend only on the smooth structure and are related to complex structure when the manifold supports Kähler metric. This is no more true for manifolds with complex dimension greater than two. Attempts, based on information of string theory, have been to use symplectic geometry to replace such invariants. This includes the so-called Gromov-Witten invariants. They are about counting curves in a manifold. The theory is pretty complete for curves of genus zero. But it is much more difficult to handle curves of higher genus. 56 / 70

57 But it is also desirable to count other important objects such as Hermitian Yang-Mills bundles and special Lagrangian cycles. In connection with this, the Hodge conjecture stands out to be most prominent conjecture which can only be true for rational (p, p)-classes of projective manifolds. There is little progress on the Hodge conjecture besides the cases of dimension 1 and codimension 1 (Lefschetz theorem). A theorem of Chern (1954) says that Chern classes of holomorphic bundles are representable by algebraic cycles. One approach is to try to prove that (p, p)-closed forms can be represented by Chern classes of some holomorphic bundles. But that is not easy. 57 / 70

58 The formulation of Hodge conjecture for non-algebraic Kähler manifolds is not so clear. An important problem is that in general, Kähler manifolds admit no holomorphic cycles. Several attempts have been made to enlarge the category of holomorphic cycles. Claire Voisin proposed to replace holomorphic cycles by Chern classes of holomorphic coherent sheaves. But she found an example of 4-dimensional complex torus which possesses a nontrivial Hodge class of degree 4 such that c 2 (F) = 0 for any holomorphic coherent sheaf F. 58 / 70

59 Perhaps one can represent Hodge class of degree d by the immersion of a complete Kähler manifold C with finite volume growth such that 1. The map is globally Lipschitz. 2. The self-intersection set of the immersion has measure zero with respect to the 2d Hausdorff measure. 3. At each embedded point in the image, there is a neighborhood where the image of C within this neighborhood can be written as a disjoint union of disks whose transverse measure is finite. Such objects should be interesting for Kähler geometry and there should be abundant such generalized cycles. 59 / 70

60 In particular, when the complex manifold is not hyperbolic in the sense of Kobayashi, Brody constructed a map from the complex line into the manifold. It will be nice to prove the Brody map carried further information so that a Hodge class of degree 2 exists on the manifold. For projective manifolds, one hopes that such noncompact cycles can be perturbed to be close up as a compact holomorphic cycle. This may give a way to resolve the Hodge conjecture 60 / 70

61 In recent years, there have been much attempts to study Kähler geometry over p-adic numbers. This can be a very promising way to understand Kähler geometry itself. Note that several important results in Kähler geometry are based on methods from arithmetic and algebraic geometry. The most notable is Mori s method of bend and break on the construction of rational curves for manifolds with numerical positive first Chern class. This spectacular result had not been recovered by standard methods in Kähler geometry. Another result is the work of Bloch-Gieseker on the positivity of m-th Chern class of rank m bundle over a projective manifold with dimension m. This was proved by hard Lefschetz theorem. It is not clear whether one can prove it by curvature representation of the Chern form. 61 / 70

62 The difficulty in understanding non-projective Kähler geometry is that it is difficult to construct holomorphic objects such as holomorphic cycles or holomorphic coherent sheaves in such geometry. However, in some situations, non-existence of holomorphic objects can be a blessing as it may mean that obstructions for the existence of some canonical geometric quantity disappear. 62 / 70

63 About twenty years ago, by using the fact that for class VII 0 surfaces with no holomorphic curves, Jun Li, Fangyang Zheng and I can construct Hermitian Yang-Mills connection on the tangent bundle to settle the classification for such surfaces. The key idea is that non-existence of holomorphic curves allows us to prove stability of the tangent bundle. This is a good example to convert an inconvenient condition to one which is favorable. We hope that this can be used more often for treating non-algebraic Kähler manifolds. 63 / 70

64 It was a well known conjecture of Kodaira that any compact Kähler manifold can be deformed to an algebraic manifold. About ten years ago, Voisin constructed examples of Kähler manifolds, no smooth bimeromorphic model of which are homotopic to an algebraic manifold. Hence Kähler manifolds form a substantially larger category than algebraic manifolds. 64 / 70

65 Many powerful methods in Kähler geometry come from algebraic geometry. However, there are also transcendental methods that may not be easily replaced by algebraic methods. Besides Kähler metrics with constant scalar curvature mentioned above, there are works related to Donaldson-Uhlenbeck-Yau theorem on the existence of Hermitian Yang-Mills connections on polystable holomorphic bundles. A very important consequence of such connections is that under merely some vanishing conditions on the wedge product of Kähler classes with first and second Chern classes, one concludes that the bundle is projective flat! It gives a vast generalization of the theorem of Narasimhan and Seshadri for bundles over curves. 65 / 70

66 There are noncompact version of the theorem of DUY, but it is not general enough to cover all cases; it would be useful to give necessary and sufficient conditions for bundles over a complete Kähler-Einstein manifold to admit Hermitian Yang-Mills connections. It is important to understand the Chern forms defined by the curvature of the Hermitian Yang-Mills connection. This is especially important for the first and second Chern forms. We would like to see the difference of Chern forms of the bundle with the corresponding Chern forms of the base metric to have nice behavior near infinity, e.g., wedge with the Kähler class becomes integrable. 66 / 70

67 In physics, it is natural to couple gravity, gauge theory and scalar field together. But it is not so easy to construct solutions without the presence of enough continuous symmetries. A natural question is to do it in complex geometry and hope that one can reduce the problem to manageable nonlinear equations. A natural approach was based on the work of Andrew Strominger. He provided a system of four equations for a supersymmetric vacuum solution for heterotic string theory. It is natural that one has to give up Kählerness of the metric. He demands the metric to be Hermitian and balanced in the sense of Michelson. 67 / 70

68 Besides the Hermitian Yang-Mills equations, there is an equation called anomaly equation that linked bundle theory with balanced metrics. These equations are not easy to solve at all. The first nontrivial solution was found by Fu and myself nine years ago. Such theory of non-kähler manifolds is very beautiful although much more needs to be done. In the past few years, LeBrun and his coauthors considered the complex surface case when the metric is conformally Kähler and the gauge group is abelian. He constructed many beautiful examples and showed that we should explore much more in this direction. 68 / 70

69 I proposed to use harmonic map to study rigidity of complex structure, which I introduced to Siu to study rigidity of compact Kähler manifolds with negative curvature. There are works related to determine the structure of complete manifolds according to signs of curvature, e.g., on Frenkel conjecture and my conjecture that complete Kähler manifolds with positive sectional curvature is C n. There are also ideas related to compactification of complete Kähler manifolds initiated in works of Siu-Yau and Mok. Many of the above programs are linked together in different forms through metric geometry, algebraic manifolds and physics. Ideas from all these disciplines are very important. But a fundamental fact remains: good estimates of quantities in the equations that govern the geometric structure! It is often dangerous to avoid such estimates. On the other hand, it is rewarding to find a good estimate to build a geometric structure that is useful. 69 / 70

70 Thank you! 70 / 70