Generalized Kummer Varieties Research Articles

Absolutely trianalytic tori in the generalized Kummer variety

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.

A Proof of the Kawamata–Morrison Cone Conjecture for Holomorphic Symplectic Varieties ofK3[n]or Generalized Kummer Deformation Type

We prove a version of the Kawamata-Morrison ample cone conjecture for projective irreducible holomorphic symplectic manifolds deformation equivalent to either the Hilbert scheme of n points on a K3 surface, or a generalized Kummer variety.

On Derived Autoequivalences of Hilbert Schemes and Generalized Kummer Varieties

On Derived Autoequivalences of Hilbert Schemes and Generalized Kummer Varieties

Derived autoequivalences of generalised Kummer varieties

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.

Beauville-Voisin Conjecture for Generalized Kummer Varieties

Inspired by their results on the Chow rings of projective K3 surfaces, Beauville and Voisin made the following conjecture: given a projective hyperkähler manifold, for any algebraic cycle that is a polynomial with rational coefficients of Chern classes of the tangent bundle and line bundles, it is rationally equivalent to zero if and only if it is numerically equivalent to zero. In this paper, we prove the Beauville–Voisin conjecture for generalized Kummer varieties.

Fourier-Mukai Partners of Canonical Covers of Bielliptic and Enriques Surfaces

We prove that the canonical cover of an Enriques surface does not admit non-trivial Fourier-Mukai partners. We also show that the canonical cover of a bielliptic surface has at most one non-isomorphic Fourier-Mukai partner. The first result is then applied to birational Hilbert schemes of points and the second to birational generalised Kummer varieties

Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties

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 Kummer varieties that come from an automorphism of the underlying abelian surface.

Intersection Numbers of Extremal Rays on Holomorphic Symplectic Varieties

We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.

Twisted cohomology of the Hilbert schemes of points on surfaces

We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in local systems. For that purpose, we generalise I. Grojnoswki's and H. Nakajima's description of the ordinary cohomology in terms of a Fock space representation to the twisted case. We make further non-trivial generalisations of M. Lehn's work on the action of the Virasoro algebra to the twisted and the non-projective case. Building on work by M. Lehn and Ch. Sorger, we then give an explicit description of the cup-product in the twisted case whenever the surface has a numerically trivial canonical divisor. We formulate our results in a way that they apply to the projective and non-projective case in equal measure. As an application of our methods, we give explicit models for the cohomology rings of the generalised Kummer varieties and of a series of certain even dimensional Calabi–Yau manifolds.

The Hilbert scheme of points for supersingular abelian surfaces

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.

The moduli space of hyper-Kähler four-fold compactifications

I discuss some aspects of the moduli space of hyper-K{\"a}hler four-fold compactifications of type II and ${\cal M}$- theories. The dimension of the moduli space of these theories is strictly bounded from above. As an example I study Hilb$^2(K3)$ and the generalized Kummer variety $K^2(T^4)$. In both cases RR-flux (or $G$-flux in ${\cal M}$-theory) must be turned on, and we show that they give rise to vacua with ${\cal N}=2$ or ${\cal N}=3$ supersymmetry upon turning on appropriate fluxes. An interesting subtlety involving the symmetric product limit $S^2(K3)$ is pointed out.

Computing the Euler characteristic of generalized Kummer varieties

We give an elementary proof of the formula χ(KnA)=n3σ(n) for the Euler characteristic of the generalized Kummer variety KnA, where σ(n) denotes the sum of divisors function.

Lagrangian fibrations on generalized Kummer varieties

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 equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K n A carries a divisor with vanishing Beauville-Bogomolov square.

On the elliptic genus of generalised Kummer varieties

Borisov and Libgober ([2]) recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde (see [6]) on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of generalised Kummer varieties [17] to deduce the following formula, conjectured by Kawai and Yoshioka ([15]), on the elliptic genus of a generalised Kummer variety A[[ n >]] of dimension 2(n−1): Here is the weak Jacobi form of weight −1 and index and V(n) is the Hecke operator sending Jacobi forms of index r to Jacobi forms of index nr (see [7]).

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in P 3g−3 for generic curves), described by 2 Θ functions or second logarithmic derivatives of theta. We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.

Calculation of Rozansky–Witten invariants on the Hilbert schemes of points on a K3 surface and the generalised Kummer varieties

For any holomorphic symplectic manifold (X,¾), a closed Jacobi diagram with 2k trivalent vertices gives rise to a Rozansky- Witten class RWX,¾(i) 2 H 2k (X,OX). If X is irreducible, this defines a number ¯i(X,¾) by RWX,¾(i) = ¯i(X,¾)(¯¾) k .

On the Chern numbers of generalised Kummer varieties

On the Chern numbers of generalised Kummer varieties

On the Euler characteristic of generalized Kummer varieties

We apply the Yau-Zaslow-Beauville method to compute the Euler characteristic of the generalized Kummer varieties attached to a complex abelian surface A (a calculation also done by Göttsche and Soergel by a different method). The argument is very geometric: given a nondivisible ample line bundle L of degree n on A , we construct a projective symplectic variety J with a Lagrangian fibration whose Euler characteristic is n times the number of genus 2 curves in | L |, to wit n 2 Σ m | n m . Using the Mukai-Fourier transform, a degeneration to the case when A is a product of elliptic curves and a result of Huybrechts, we prove that J and the generalized (2 n -2)-dimensional Kummer variety of A are diffeomorphic, hence have the same Euler characteristic.

Polarized Varieties, Fields of Moduli and Generalized Kummer Varieties of Polarized Abelian Varieties

Polarized Varieties, Fields of Moduli and Generalized Kummer Varieties of Polarized Abelian Varieties

Polarized varieties, the fields of moduli and generalized Kummer varieties of abelian varieties

Polarized varieties, the fields of moduli and generalized Kummer varieties of abelian varieties