Moduli spaces of sheaves on K3 surfaces and Galois representations

Frei, Sarah

doi:10.1007/s00029-019-0530-7

Cited by 10 publications

(8 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of moduli spaces M H (v) of Gieseker-stable sheaves, with Mukai vector v, with respect to a generic polarization H on the K3 surface S (which are special cases of moduli spaces of σ-stable objects in D b (S)), Markman [Mar08,§3.4] has established that there exists an isomorphism between the cohomology algebras of M H (v) and Hilb n (S) that in addition preserves the Hodge structures. Recently, Frei [Fre20] extended Markman's result to positive characteristic, with ℓ-adic cohomology with its Galois structure, in place of singular cohomology with its Hodge structure.…”

Section: The Birational Motive Of Moduli Spaces Of Sheaves On K3 Surf...mentioning

confidence: 99%

On the birational motive of hyper-Kähler varieties

Vial¹

2020

Preprint

View full text Add to dashboard Cite

We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for certain moduli spaces of stable sheaves on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with respect to the diagonal embedding and to show in the aforementioned cases the existence of a so-called strict grading. Furthermore, we upgrade to rational equivalence Voisin's notion of "surface decomposition" and use this to show that the birational motive of some projective hyper-Kähler manifolds is determined, as a co-algebra object, by the birational motive of a surface. In an appendix, we relate our co-radical filtration on the Chow groups of abelian varieties to Beauville's eigenspace decomposition.

show abstract

Section: The Birational Motive Of Moduli Spaces Of Sheaves On K3 Surf...mentioning

confidence: 99%

On the birational motive of hyper-Kähler varieties

Vial¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Theorem 3 stands in contrast to a result of Frei [Fre20, Theorem 1], who showed that, over a finite field, two smooth projective moduli spaces of sheaves on a given K3 surface have the same number of points as soon as they have the same dimension.…”

Section: Introductionmentioning

confidence: 76%

“…The usual identification of with the orthogonal to the Mukai vector in the Mukai lattice is valid as -modules; to see this, note that all the maps appearing in [Cha16, Theorem 2.4(vi)] are -equivariant, and see [Fre20, § 2] for the techniques needed to relax condition (C) of [Cha16, Definition 2.3].…”

Section: A Hyperkähler Counterexamplementioning

confidence: 99%

Rational points and derived equivalence

Addington¹,

Antieau²,

Frei³

et al. 2021

Compositio Math.

Self Cite

View full text Add to dashboard Cite

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

show abstract

“…In particular Z 1,k and Z 2,k have the same zeta function. In the special case when Z 1 and Z 2 are moduli spaces of stable sheaves on K3 surfaces over k the above statement has already been proven by Frei in [13] via a different method, which uses Markman's results from [19].…”

Section: Introductionmentioning

confidence: 83%

Galois representations on the cohomology of hyper-Kähler varieties

Floccari

2022

Math. Z.

View full text Add to dashboard Cite

We show that the André motive of a hyper-Kähler variety X over a field $$K \subset {\mathbb {C}}$$ K ⊂ C with $$b_2(X)>6$$ b 2 ( X ) > 6 is governed by its component in degree 2. More precisely, we prove that if $$X_1$$ X 1 and $$X_2$$ X 2 are deformation equivalent hyper-Kähler varieties with $$b_2(X_i)>6$$ b 2 ( X i ) > 6 and if there exists a Hodge isometry $$f:H^2(X_1,{\mathbb {Q}})\rightarrow H^2(X_2,{\mathbb {Q}})$$ f : H 2 ( X 1 , Q ) → H 2 ( X 2 , Q ) , then the André motives of $$X_1$$ X 1 and $$X_2$$ 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 $$X_1$$ X 1 and $$X_2$$ 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.

show abstract

Moduli spaces of sheaves on K3 surfaces and Galois representations

Cited by 10 publications

References 27 publications

On the birational motive of hyper-Kähler varieties

On the birational motive of hyper-Kähler varieties

Rational points and derived equivalence

Galois representations on the cohomology of hyper-Kähler varieties

Contact Info

Product

Resources

About