Related papers: Derived equivalences of generalized Kummer varieti…

In this article, we study the $G$-autoequivalences of the derived category $\mathbf{D}^b_G(A)$ of $G$-equivariant objects for an abelian variety $A$ with $G$ being a finite subgroup of $\mathrm{Pic}^0(A)$. We provide a result analogue to…

In this article, we construct new derived autoequivalences of generalised Kummer varieties. Together with Huybrechts-Thomas twists around $\mathbb{P}^n$-objects, these are the only known examples of such symmetries.

We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary…

Over complex numbers, the Fourier-Mukai partners of abelian varieties are well-understood. A celebrated result is Orlov's derived Torelli theorem. In this note, we study the FM-partners of abelian varieties in positive characteristic. We…

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup…

We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K^n(A) is birationally…

We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$ disjoint…

We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized…

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 present a method to construct explicit degenerations of higher-dimensional generalized Kummer varieties. We start with a simple degeneration $f: \mathcal Y \to C$ of abelian surfaces. Then $ \mathcal{Y} \setminus \mathcal{Y}_0$ is an…

We apply the Yau-Zaslow-Beauville method to compute the Euler characteristic of the generalized Kummer varieties attached to a complex abelian surface (a calculation also done by Goettsche and Soergel by different methods). It is related to…

Given an abelian variety $A$ over a number field, we consider the generalized Kummer varieties of $A$ coming from quotients of $A$ by an automorphism of prime order $p > 2$. We prove that the Brauer-Manin obstruction on these generalized…

We discuss several geometric features of a Kummer surface associated with a (1,2)-polarized abelian surface defined over the field of complex numbers. In particular, we show that any such Kummer surface can be modeled as the double cover of…

We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we…

We show that for every smooth projective surface X and every $n\ge 2$ the push-forward along the diagonal embedding gives a $P^{n-1}$-functor into the $S_n$-equivariant derived category of X^n. Using the Bridgeland--King--Reid--Haiman…

We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which include any abelian variety of dimension atmost 5, powers of complex elliptic curves, etc.) which vanishes as a morphism on a…

We prove general type results for orthogonal modular varieties associated with the moduli of compact hyperk\"ahler manifolds of deformation generalised Kummer type ('deformation generalised Kummer varieties'). In particular, we consider…

We compute the Euler characteristics of the generalized Kummer schemes associated to $A\times Y$, where $A$ is an abelian variety and $Y$ is a smooth quasi-projective variety. When $Y$ is a point, our results prove a formula conjectured by…

We determine normal forms for the Kummer surfaces associated with abelian surfaces of polarization of type $(1,1)$, $(1,2)$, $(2,2)$, $(2,4)$, and $(1,4)$. Explicit formulas for coordinates and moduli parameters in terms of Theta functions…

A generalized Kummer surface $X=Km(T,G)$ is the resolution of a quotient of a torus $T$ by a finite group of symplectic automorphisms $G$. We complete the classification of generalized Kummer surfaces by studying the two last groups which…