On the coniveau of certain sub-Hodge structures

We study the generalized Hodge conjecture for certain sub-Hodge structure defined as the kernel of the cup product map with a big cohomology class, which is of Hodge coniveau at least 1. As predicted by the generalized Hodge conjecture, we prove that the kernel is supported on a divisor, assuming the Lefschetz standard conjecture.Comment: 23 pages. V2: Typos corrected. Comments still welcome. To appear in Math.Res.Let

Beauville-Voisin conjecture for generalized Kummer varieties

Inspired by their results on the Chow rings of projective K3 surfaces, Beauville and Voisin made the following conjecture: given a projective hyperkaehler manifold, for any algebraic cycle which is a polynomial with rational coefficients of Chern classes of the tangent bundle and line bundles, it is rationally equivalent to zero if and only if it is numerically equivalent to zero. In this paper, we prove the Beauville-Voisin conjecture for generalized Kummer varieties.Comment: 14 pages. v2: Last section expanded. Published online in International Mathematics Research Notices 201

Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi-Yau complete intersections

On one hand, for a general Calabi-Yau complete intersection X, we establish a decomposition, up to rational equivalence, of the small diagonal in X^3, from which we deduce that any decomposable 0-cycle of degree 0 is in fact rationally equivalent to 0, up to torsion. On the other hand, we find a similar decomposition of the smallest diagonal in a higher power of a hypersurface, which provides us an analogous result on the multiplicative structure of its Chow ring.Comment: 33 pages. Comments are welcom

CLASSIFICATION OF POLARIZED SYMPLECTIC AUTOMORPHISMS OF FANO VARIETIES OF CUBIC FOURFOLDS

International audienc

Motivic integration and the birational invariance of BCOV invariants

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called the BCOV invariant. The BCOV invariant is conjecturally related to the Gromov-Witten theory via mirror symmetry. Based upon previous work of the second author, we prove the conjecture that birational Calabi-Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant, as well as its birational invariance, to Calabi-Yau varieties with Kawamata log-terminal singularities. We also give an interpretation of our construction using the theory of motivic integration.Comment: 31 pages. Comments welcome

Algebraic cycles on Gushel-Mukai varieties

We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic.Comment: 24 page

Finiteness of Klein actions and real structures on compact hyperkähler manifolds

open2One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperkähler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperkähler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperkähler manifold is finitely presented.openCattaneo A.; Fu L.Cattaneo, A.; Fu, L

Cubic fourfolds, Kuznetsov components and Chow motives

We prove that the Chow motives of two smooth cubic fourfolds whose Kuznetsov components are Fourier-Mukai derived-equivalent are isomorphic as Frobenius algebra objects. As a corollary, we obtain that there exists a Galois-equivariant isomorphism between their l-adic cohomology Frobenius algebras. We also discuss the case where the Kuznetsov component of a smooth cubic fourfold is Fourier-Mukai derived-equivalent to a K3 surface.Comment: 21 page