Abstract
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 (semi-)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 7.3 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.
Highlights
An important source of constructions of higher-dimensional algebraic varieties is given by taking moduli spaces of coherent sheaves, subject to various stability conditions, on some lower-dimensional algebraic varieties
The topological, geometric, algebraic and arithmetic properties of the variety are certainly expected to be reflected in and sometimes even control the corresponding properties of the moduli space. Such relations can be made precise in terms of cohomology groups or more fundamentally, at the level of motives.a The prototype of such interplay we have in mind is del Bano’s result [27], which says that the Chow motive of the moduli space Mr,d(C) of stable vector bundles of coprime rank and degree on a smooth projective curve C is a direct summand of the Chow motive of some power of the curve; in other words, the Chow motive of Mr,d(C) is in the pseudo-abelian tensor subcategory generated by the Chow motive of C
As is alluded to in the introduction, many projective hyper-Kahler varieties are constructed as moduli spaces of objects in some 2-Calabi–Yau categories, and it is natural to wonder how the motive of the moduli space is related to the “motive” of this category, whatever it means.g gThe motive of a differential graded category can certainly be made precise: it is the theory of non-commutative motives, see Tabuada [82] for a recent account
Summary
An important source of constructions of higher-dimensional algebraic varieties is given by taking moduli spaces of (complexes of) coherent sheaves, subject to various stability conditions, on some lower-dimensional algebraic varieties. The topological, geometric, algebraic and arithmetic properties of the variety are certainly expected to be reflected in and sometimes even control the corresponding properties of the moduli space Such relations can be made precise in terms of cohomology groups (enriched with Hodge structures and Galois actions for instance) or more fundamentally, at the level of motives.a The prototype of such interplay we have in mind is del Bano’s result [27], which says that the Chow motive of the moduli space Mr,d(C) of stable vector bundles of coprime rank and degree on a smooth projective curve C is a direct summand of the Chow motive of some power of the curve; in other words, the Chow motive of Mr,d(C) is in the pseudo-abelian tensor subcategory generated by the Chow motive of C. It is worth pointing out that Bulles’ method gives a short and new proof of del Bano’s result using the classical analogue of Markman’s result in the curve case proved by Beauville [13]
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