Japan. J. Math. 2, 261–296 (2007) DOI: 10.1007/s11537-007-0639-x Some aspects of the Hodge conjecture Claire Voisin Received: 17 February 2007 / Revised: 19 May 2007 / Accepted: 19 May 2007 Published online: 21 September 2007 c The Mathematical Society of Japan and Springer 2007 Communicated by: Hiraku Nakajima Abstract. I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the other side from the study of possible extensions of the conjecture to the general K¨ahler setting. The positive aspects come from algebraic geometry. They concern the structure of the so-called locus of Hodge classes, and of the Hodge loci. Keywords and phrases: Hodge classes, Chern classes, Hodge loci Mathematics Subject Classification (2000): 14C30, 32J25 Contents 0. Introduction ............................................................................................ 262 0.1. Hodge theory ................................................................................... 262 0.2. Hodge classes and the Hodge conjecture ................................................. 265 0.3. Lefschetz theorem on (1,1)-classes ........................................................ 266 0.4. Organisation of the paper .................................................................... 267 1. The analytic side: Hodge classes and coherent sheaves on K¨ahler manifolds ............. 268 1.1. Several constructions of Hodge classes ................................................... 268 1.2. Weil tori and Weil classes .................................................................... 271 1.3. The main result ................................................................................ 273 1.4. Further comments on the K-theory of compact K¨ahler manifolds ................... 276 2. The topological side: integral Hodge classes .................................................... 277 2.1. Atiyah–Hirzebruch–Totaro topological obstruction ..................................... 277 This article is based on the 1st Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 25 and 26, 2006. C. VOISIN CNRS UMR 7586, Institut de Math´ematiques de Jussieu, 175 rue du Chevaleret 75103 Paris, France (e-mail: [email protected]) 262 C. Voisin 2.2. Koll´ar’sexample ............................................................................... 279 2.3. Rationally connected varieties and the rationality problem ............................ 281 3. The algebraic side: absolute Hodge classes and Hodge loci ................................... 286 3.1. Absolute Hodge classes ...................................................................... 286 3.2. Locus of Hodge classes ...................................................................... 290 0. Introduction This text is an expanded version of the Takagi lectures I gave at the RIMS, Kyoto, in November 2006. Following my work of the last years around the Hodge conjecture (see [38], [40], and [41]), I decided to insist in these lectures on the fact that, while the Hodge conjecture can be considered as a conjecture either in complex analysis, in differential topology, or in algebraic geometry, the evidences for it are all coming from algebraic geometry. In fact, the paper [38] shows that, if one extends the geometric setting and consider the case of compact Kahler¨ manifolds, one finds supplementary re- strictions for Hodge classes to come from constructions of complex analytic geometry. On the other hand, in the context of algebraic geometry, one can ask ques- tions concerning the definition fields of the Hodge loci, as the Hodge conjecture predicts that they are algebraic and defined on Q (here we note that any complex projective manifold is a member of a family defined over Q). The best evidence for the Hodge conjecture is thus due to Cattani, Deligne and Kaplan [10], who showed that the Hodge loci, and even the connected com- ponents of the locus of Hodge classes, are algebraic. We provide in [41] an improvement of their result by giving some criteria for the Hodge loci to be defined over Q. We also relate this problem to the question whether the Hodge conjecture can be reduced to the case of varieties defined over Q. We included in the text a section devoted to the case of integral Hodge classes, for which the original Hodge conjecture was disproved by Atiyah and Hirzebruch [4]. We discuss there the possibility of extracting interesting bira- tional invariants from this, using the groups of integral Hodge classes modulo those which are algebraic. This is essentially taken from [40], but the discussion concerning the 4-dimensional case here is new. Acknowledgements. I thank Yasuyuki Kawahigashi, Toshiyuki Kobayashi, Hiraku Nakajima, Kaoru Ono and Takeshi Saito, the organizers of the first Takagi lectures, for inviting me to deliver these lectures and for giving me the opportunity to write these notes. 0.1. Hodge theory Let us start with the notion of cohomology class of type (p,q) on a complex manifold X.OnsuchanX, we have the notion of differential form of type Some aspects of the Hodge conjecture 263 ∞ (p,q): these are the complex differential forms ω (say of class ), which can be written in local holomorphic coordinates z1,...,zn, n = dimCX, and in the multiindex notation: ω = ∑ωI,J dzI ∧ dzJ, |I| = p, |J| = q, I,J ∞ p,q where ωI,J are C functions. Let us denote by A (X) the space of (p,q)-forms. Thus Ap,q(X) ⊂ Ak(X), p + q = k,whereAk(X) is the space of C∞ complex differential k-forms on X. Recalling that the complex de Rham cohomology Hk(X,C) is defined as {closed complex k-forms on X} Hk(X,C)= , {exact complex k-forms on X} it is natural to define for a complex manifold X and for each (p,q) the space of cohomology classes of type (p,q) by the formula {closed complex forms of type (p,q) on X} H p,q(X) := . {exact forms of type (p,q) on X} Each complex differential form of degree k decomposes canonically into its components of type (p,q) because any complex differential form of degree 1 decomposes canonically as a sum of a form of type (1,0) (its C-linear part) and a form of type (0,1) (its C-antilinear part). Unfortunately, the differential d does not preserve this decomposition: in fact, we have d = ∂ + ∂, where ∂ : Ap,q(X) → Ap+1,q(X),and∂ : Ap,q(X) → Ap,q+1(X). For this reason, the notion of class of type (p,q) is not very useful for general complex mani- folds. First of all, there might be no non-zero class of type (p,q),forany(p,q) such that p+q = k, while Hk(X,C) = 0. The standard example is that of a Hopf surface, and k = 1. Furthermore, it might be the case that a non-zero cohomol- ogy class could be both of type (p,q) and (p,q) with (p,q) =(p,q). These pathologies do not appear in the compact Kahler¨ case.Thisisafa- mous result due to Hodge, which comes as a consequence of the theory of har- monic forms, and of the so-called Kahler¨ identities. AKahler¨ metric is a Hermitian metric on the tangent bundle of a complex manifold X (which has a natural structure of complex vector bundle), which fits very nicely with the complex structure on X. The Hermitian metric h be- ⊗ ing locally written in holomorphic coordinates as ∑i, j hijdzi dz j, there is the corresponding real (1,1)-form i ω = ∑hijdzi ∧ dz j, 2 i, j 264 C. Voisin (the Kahler¨ form), and the Kahler¨ condition is simply dω = 0. One good way to see that these metrics fit nicely with the complex structure is the following characterization: Proposition 1. The metric h on X is Kahler¨ if and only if the operator of com- plex structure I (defining the complex structure of the tangent bundle of X) is parallel with respect to the Levi–Civita connection associated to the underlying Riemannian metric g. From our point of view, the importance of the Kahler¨ condition lies in the fol- lowing theorem, due to Hodge. ∗ ∗ Theorem 2. If (X,ω) is Kahler,¨ the Laplacian Δg = dd +d d associated to the underlying metric g is bihomogeneous, namely satisfies p,q p,q Δg(A (X)) ⊂ A (X). Recall that a form α is said to be harmonic if Δgα = 0, or equivalently in the compact case, dα = d∗α = 0. Corollary 3. A differential form α on a Kahler¨ manifold X is harmonic if and only if its components of type (p,q) are harmonic. Note that, in the compact case, α and its components α p,q are closed, and this provides a decomposition of the cohomology class of α into classes of type (p,q). One other remarkable fact, which implies Theorem 2, is the fact that we have ∗ ∗ the following relation between the three Laplacians Δg, Δ∂ := ∂∂ +∂ ∂, Δ∂ := ∗ ∗ ∂∂ + ∂ ∂: Δg = 2Δ∂ = 2Δ∂ . As a consequence of these facts, one gets the famous Hodge decomposition theorem: Theorem 4 (The Hodge decomposition theorem). If a complex manifold X is compact and Kahler¨ (that is, admits a Kahler¨ metric), then for any integer k, there is a (natural) decomposition into a direct sum of complex subspaces Hk(X,C)= H p,q(X), p+q=k where H p,q(X) ⊂ Hk(X,C) was defined above. This theorem uses the representation of cohomology classes by harmonic forms, together with the decomposition of harmonic forms. It is a remarkable fact that the final decomposition on cohomology is in fact metric independent. Some aspects of the Hodge conjecture 265 —Hodge symmetry. It is clear by definition that H p,q(X)=Hq,p(X).This is because the complex conjugate of a closed form of type (p,q) isaclosed form of type (q, p). Here the complex conjugation acts on Hk(X,C) which is the complexification of the real cohomology space Hk(X,R). In the sequel, we will rather consider the Hodge filtration defined by FlHk(X,C)= H p,k−p(X). (0.1) p≥l This data is equivalent to the data of the Hodge decomposition, because by Hodge symmetry, we recover H p,q(X) as H p,q(X)=F pH p+q(X,C) ∩ FqH p+q(X,C), but it has the great advantage over the Hodge decomposition that it varies holo- morphically, and even algebraically (see section 3) with the complex structure.