Related papers: Footnotes to papers of O'Grady and Markman

We prove that any hyper-K\"{a}hler 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…

With any hyper-K\"ahler 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…

We prove the conjectures of Hodge and Tate for any six-dimensional hyper-K\"ahler variety that is deformation equivalent to a generalized Kummer variety.

We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperk\"ahler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove…

We prove that the natural representaion, on the total Betti cohomolgy group, of the automorphism group of a hyperk\"ahler manifold deformation equivalent to a generalized Kummer manifold is faithful. This is a sort of generalization of an…

We study the Mumford--Tate conjecture for hyperk\"{a}hler 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…

We prove the conjectures of Hodge and Tate for any four-dimensional hyper-K\"ahler 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…

We prove the formality and the evenness of odd-degree Betti numbers for compact K\"ahler orbifolds, by adapting the classical proofs for K\"ahler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any…

Let X and Y be compact hyper-Kahler manifolds deformation equivalence to the Hilbert scheme of length n subschemes of a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subring…

We address the problem of classification of hyper-K\"ahler fourfolds with $b_2=23$. In particular we prove some special cases of the Conjecture of O'Grady about hyper-K\"ahler $4$-folds numerically equivalent to the Hilbert scheme of two…

We use the extended Mukai vectors for hyper-K\"ahler manifolds to investigate the derived equivalences of the hyper-K\"ahler manifolds which are deformation equivalent to generalized Kummer varieties. Inspired by the idea for hyper-K\"ahler…

We show that a moduli space of slope-stable sheaves over a K3 surface is an irreducible hyperk\"ahler manifold if and only if its second Betti number is the sum of its Hodge numbers $h^{2,0}$, $h^{1,1}$ and $h^{0,2}$.

Let $X$ be a compact K\"ahler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X'$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, \Omega^2)…

We prove that a hyper-K\"ahler fourfold satisfying a mild topological assumption is of K3$^{[2]}$ deformation type. This proves in particular a conjecture of O'Grady stating that hyper-K\"ahler fourfolds of K3$^{[2]}$ numerical type are of…

In this paper we survey geometric and arithmetic techniques to study the cohomology of semiprojective hyperkaehler manifolds including toric hyperkaehler varieties, Nakajima quiver varieties and moduli spaces of Higgs bundles on Riemann…

We prove that the complex cobordism class of any hyper-K\"{a}hler manifold of dimension $2n$ is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of $K3$ surfaces. We also prove a similar…

We introduce the notion of a Hyper-K\"{a}hler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-K\"{a}hler manifolds studying those that are induced by a K3 or abelian…

We extend several $g$-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain K\"uhnel-type bounds on the Betti numbers as well as on certain…

In arXiv:1710.01672, we obtained general type results for orthogonal modular varieties associated with moduli spaces of compact hyperk\"ahler manifolds of deformation generalised Kummer type (also known as 'deformation generalised Kummer…

We generalize to nearly K\"ahler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly K\"ahler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly K\"ahler manifold of any…