On the Mumford–Tate conjecture for hyperkähler varieties

@article{Floccari2019OnTM,
  title={On the Mumford–Tate conjecture for hyperk{\"a}hler varieties},
  author={Salvatore Floccari},
  journal={arXiv: Algebraic Geometry},
  year={2019}
}
We study the Mumford-Tate conjecture for hyperkahler varieties. Building on work of Markman, we show that it holds in arbitrary codimension for all varieties of $\mathrm{K}3^{[m]}$-type. For an arbitrary hyperkahler variety satisfying $b_2(X)>3$ we establish one of the two inclusions of algebraic groups predicted by the Mumford-Tate conjecture. Our results extend a theorem of Andre. 
Deformation Principle and André motives of Projective Hyperkähler Manifolds
  • A. Soldatenkov
  • Mathematics
  • International Mathematics Research Notices
  • 2021
Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying thisExpand
On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties
We investigate how the motive of hyper-K\"ahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singularExpand

References

SHOWING 1-10 OF 20 REFERENCES
On the Tate and Mumford–Tate conjectures in codimension 1 for varieties with h2,0 = 1
We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0,Expand
On the Shafarevich and Tate conjectures for hyperkähler varieties
1. Problems and results 2. Polarized hyperk~ihler varieties and cubic fourfoids 3. The period mapping 4. The Kuga-Satake construction 5. The Kuga-Satake construction in the relative setting 6. ProofExpand
Kuga-Satake construction and cohomology of hyperkähler manifolds
Abstract Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H 2 ( M , C ) into the second cohomology of a torus, compatible with the Hodge structure. We constructExpand
Deformation Principle and André motives of Projective Hyperkähler Manifolds
  • A. Soldatenkov
  • Mathematics
  • International Mathematics Research Notices
  • 2021
Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying thisExpand
Compact hyperkähler manifolds: basic results
The paper generalizes some of the well-known results for K3 surfaces to higher-dimensional irreducible symplectic (or, equivalently, compact irreducible hyperkaehler) manifolds. In particular, weExpand
Families of Motives and the Mumford–Tate Conjecture
We give an overview of some results and techniques related to the Mumford–Tate conjecture for motives over finitely generated fields of characteristic 0. In particular, we explain how working inExpand
Desingularized moduli spaces of sheaves on a K3, I
Moduli spaces of semistable torsion-free sheaves on a K3 surface $X$ are often holomorphic symplectic varieties, deformation equivalent to a Hilbert scheme parametrizing zero-dimensional subschemesExpand
A Lie algebra attached to a projective variety
A bstract . Each choice of a Kähler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such Kähler classes is given,Expand
Cohomology of compact hyperkähler manifolds and its applications
We announce the structure theorem for theH2(M)-generated part of cohomology of a compact hyperkähler manifold. This computation uses an action of the Lie algebra so(4,n−2) wheren=dimH2(M) on theExpand
Hodge Cycles on Abelian Varieties
The main result proved in these notes is that any Hodge cycle on an abelian variety (in characteristic zero) is an absolute Hodge cycle — see §2 for definitions and (2.11) for a precise statement ofExpand
...
1
2
...