On the Hodge conjecture for hyperkähler manifolds

Abstract

The Hodge conjecture predicts a deep connection between topology, complex geometry, and algebraic geometry. It asserts that any Hodge class on a smooth complex projective variety is algebraic, meaning it is a linear combination with rational coefficients of fundamental classes of subvarieties. <br /> For powers of a K3 surface, the Hodge conjecture is equivalent to the alge- braicity of the Hodge classes in the tensor algebra of its <em>transcendental lattice</em>, which is the orthogonal complement of the Néron–Severi group in the second rational cohomology. In Appendix B, using invariant theory, we determine generators for this algebra of Hodge classes. By proving the existence of exceptional Hodge classes, we show that the Hodge conjecture for the powers of a K3 surface does not always follow from the Hodge conjecture for its square. Additionally, we establish the Hodge conjecture for all powers of certain K3 surfaces of Picard number 16. <br /> In Appendix A, in collaboration with Claire Voisin, we review the Kuga–Satake construction, which provides an embedding of the transcendental lattice T(X) of a projective K3 surface or hyperkähler manifold X into the square of the first cohomology of an abelian variety. If the Kuga–Satake Hodge conjecture holds, meaning this embedding is algebraic, the algebraicity of the Hodge classes in the tensor algebra of T(X) follows if one proves that their image via this embedding is algebraic on the powers of the abelian variety. Here, we implicitly assume that the Lefschetz standard conjecture in degree two holds for the variety X. This has been proven in all the cases we consider. <br /> This observation is particularly relevant for Hodge similarities between the transcendental lattices of projective hyperkähler manifolds (or K3 surfaces). These are Hodge morphisms that preserve, up to a positive scalar, the Beauville–Bogomolov forms (or the intersection products). In Appendix C, we prove that a Hodge similarity induces an isogeny between the Kuga–Satake varieties, and thus is algebraic on the product of the two Kuga–Satake varieties. If the Kuga–Satake Hodge conjecture holds, then the Hodge similarity is also algebraic on the product of the two hyperkähler manifolds (or K3 surfaces). As the Kuga–Satake Hodge conjecture has been proven for hyperkähler manifolds of generalized Kummer type, we deduce the algebraicity of any Hodge similarity between two such manifolds. This complements previous results on the algebraicity of Hodge isometries between K3 surfaces and between K3[n]-type varieties. <br /> In Appendix D, a joint work with Salvatore Floccari, we study the Hodge conjecture for hyperkähler manifolds of generalized Kummer type. We prove that any Hodge class in the algebra generated by their second cohomology is algebraic. We deduce this from three results: the algebraicity of Hodge similarities between hyperkähler manifolds of generalized Kummer type, a construction by Floccari providing an algebraic Hodge similarity between a Kum3-type variety and a K3 surface, and the fact that, by Appendix B, the Hodge conjecture holds for the K3 surfaces appearing in this construction. Remarkably, this implies the Hodge conjecture for projective Kum2-type varieties.