The Hodge and Tate conjectures for hyper-Kähler sixfolds of generalized Kummer type
We prove the conjectures of Hodge and Tate for any six-dimensional hyper-Kähler variety that is deformation equivalent to a generalized Kummer variety.
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arXiv subjects
Salvatore Floccari
Publications and source records attributed to Salvatore Floccari.
The hyper-Kummer construction
The hyper-Kummer construction, discovered by the first-named author, associates a hyper-Kähler manifold of $\mathrm{K}3^{[3]}$-type with any hyper-Kähler sixfold of generalized Kummer type. We regard this construction as a higher-dimensional analog of the classical Kummer construction of K3 surfaces from abelian surfaces. In this spirit, we prove several results which parallel those known for the classical Kummer construction: we characterize the hyper-Kummer $\mathrm{K}3^{[3]}$-manifolds up to birational equivalence in terms of their Hodge lattices, and establish a McKay correspondence for their derived categories and Chow motives. We propose a recipe to construct locally complete families of projective varieties of $\mathrm{Kum}^3$-type starting from a family of varieties of $\mathrm{K}3^{[3]}$-type equipped with 16 prime divisors in a certain Kummer lattice configuration. We also compare hyper-Kummer $\mathrm{K}3^{[3]}$-manifolds with the Mongardi-Rapagnetta-Saccà double covers of O'Grady's six-dimensional hyper-Kähler manifolds. The hyper-Kummer construction produces a rich configuration of hyper-Kähler manifolds of $\mathrm{K}3^{[2]}$-type and K3 surfaces canonically associated with a manifold of $\mathrm{Kum}^3$-type, in particular the hyper-Kummer K3 surfaces, which form countably many $4$-dimensional families of generic Picard rank 16. We prove abelianity of Chow motives for infinitely many 4-dimensional families of hyper-Kummer K3 surfaces, thereby proving Kimura-O'Sullivan finite-dimensionality conjecture for many new K3 surfaces of Picard rank 16. As other applications, we prove Beauville's weak splitting conjecture for all varieties of $\mathrm{Kum}^3$-type, and, building on previous results, we prove the Hodge and Tate conjectures for all powers of any of the varieties involved in the hyper-Kummer construction.
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The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties
We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties $\widetilde{K}$ of O'Grady type arising as crepant resolutions $\widetilde{K}\to K$ of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety $\widetilde{K}$ of OG6-type as above, and all of its powers.
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K3 surfaces associated with varieties of generalized Kummer type
With any hyper-Kähler variety $K$ of generalized Kummer type is associated via Hodge theory a K3 surface $S_K$. We show how they are related geometrically through a moduli space of sheaves on $S_K$. As a consequence, building fundamentally on the works of O'Grady, Markman, Voisin, Varesco, we establish the Hodge conjecture for all powers of any of these K3 surfaces as well as for all abelian fourfolds of Weil type with discriminant 1 and their powers, strenghtening a result of Markman.
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Algebraic cycles on hyper-Kähler varieties of generalized Kummer type
We prove the conjectures of Hodge and Tate for any four-dimensional hyper-Kähler variety of generalized Kummer type. For an arbitrary variety $X$ of generalized Kummer type, we show that all Hodge classes in the subalgebra of the rational cohomology generated by $H^2(X,\mathbb{Q})$ are algebraic.
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Sixfolds of generalized Kummer type and K3 surfaces
We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm{K}3^{[3]}$-type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective~$\mathrm{K}3$ surface~$S_K$. As application of this construction we show that the Kuga-Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm{K}3$ surfaces of general Picard rank $16$ satisfying the Kuga-Satake Hodge conjecture.
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On the motive of O'Grady's six dimensional hyper-Kähler varieties
We prove that the rational Chow motive of a six dimensional hyper-Kähler variety obtained as symplectic resolution of O'Grady type of a singular moduli space of semistable sheaves on an abelian surface $A$ belongs to the tensor category of motives generated by the motive of $A$. We in fact give a formula for the rational Chow motive of such a variety in terms of that of the surface. As a consequence, the conjectures of Hodge and Tate hold for many hyper-Kähler varieties of OG6-type.
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Galois representations on the cohomology of hyper-Kähler varieties
We show that the André motive of a hyper-Kähler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent hyper-Kähler varieties with $b_2(X_i)>6$ and if there exists a Hodge isometry $f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, then the André motives of $X_1$ and $X_2$ are isomorphic after a finite extension of $K$, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of $X_1$ and $X_2$ are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford--Tate conjecture is true.
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On the Mumford-Tate conjecture for hyperkähler varieties
We study the Mumford--Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford--Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.
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On the motive of O'Grady's ten-dimensional hyper-Kähler varieties
We investigate how the motive of hyper-Kähler 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 Bülles to the O'Grady-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-Kähler varieties of O'Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler 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-Kähler 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-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford--Tate conjecture holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.
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