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Register a new userThe generalized Franchetta conjecture for some hyper-Kahler varieties, II
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We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macr`i-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface.
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We investigate how the motive of hyper-Kahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bulles to the OGrady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kahler varieties of OGrady-10 deformation type satisfying the standard conjectures. In the second part, we study the Andr{e} motive of projective hyper-Kahler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford--Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kahler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kahler varieties, all Hodge and Tate classes are motivated, the motivated Mumford--Tate conjecture holds, and the Andre motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.
We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.
This note is an erratum to the paper Tautological classes on moduli spaces of hyper-Kahler manifolds. Thorsten Beckman and Mirko Mauri have pointed to us a gap in the proof of cite[Theorem 8.2.1]{Duke}. We do not know how to correct the proof. We can only recover a partial statement. This gap affects the proof of one of the two main results of cite{Duke}, we explain how to correct it.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
We study the generalized Kahler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We derive an extension of Perelmans entropy functional to this setting, which yields convergence of nonsingular solutions at infinity. Furthermore, we derive an extension of Mabuchis $K$-energy to this setting, which yields weak convergence of the flow.
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