Generalized Kummer Varieties Research Articles

Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds

Using reduced Gromov–Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi–Yau 3-folds, Klemm and Pandharipande for Calabi–Yau 4-folds, and Pandharipande and Zinger for Calabi–Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced 4-dimensional Donaldson–Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two K3 surfaces and for the cotangent bundle of P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a K3 surface. This yields a conjectural formula for the number of isolated genus 2 curves of minimal degree on a very general hyperkähler 4-fold of K3[2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K3^{[2]}$$\\end{document}-type. The formula may be viewed as a 4-dimensional analogue of the classical Yau–Zaslow formula concerning counts of rational curves on K3 surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.

Perverse Filtration for Generalized Kummer Varieties of Fibered Surfaces

Abstract Let $A\to C$ be a proper surjective morphism from a smooth connected quasi-projective commutative group scheme of dimension 2 to a smooth curve. The construction of generalized Kummer varieties gives a proper morphism $A^{[[n]]}\to C^{((n))}$. We show that the perverse filtration associated with this morphism is multiplicative.

Fundamental group schemes of generalized Kummer variety

Fundamental group schemes of generalized Kummer variety

On Abel–Jacobi maps of Lagrangian families

We study in this article the cohomological properties of Lagrangian families on projective hyper-Kähler manifolds. First, we give a criterion for the vanishing of Abel–Jacobi maps of Lagrangian families. Using this criterion, we show that under a natural condition, if the variation of Hodge structures on the degree 1 cohomology of the fibers of the Lagrangian family is maximal, its Abel–Jacobi map is trivial. We also construct Lagrangian families on generalized Kummer varieties whose Abel–Jacobi map is not trivial, showing that our criterion is optimal.

Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions

Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions

On the birational motive of hyper-Kähler varieties

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 moduli spaces of stable objects 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 whose associated filtration agrees with the filtration of Voisin. As results of independent interest, we upgrade to rational equivalence Voisin's notion of “surface decomposition” and 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. We also relate our co-radical filtration on the Chow groups of abelian varieties to Beauville's eigenspace decomposition.

Stable vector bundles on generalized Kummer varieties

Abstract For an abelian surface A, we explicitly construct two new families of stable vector bundles on the generalized Kummer variety K n ⁢ ( A ) {K_{n}(A)} for n ⩾ 2 {n\geqslant 2} . The first is the family of tautological bundles associated to stable bundles on A, and the second is the family of the “wrong-way” fibers of a universal family of stable bundles on the dual abelian surface A ^ {\widehat{A}} parametrized by K n ⁢ ( A ) {K_{n}(A)} . Each family exhibits a smooth connected component in the moduli space of stable bundles on K n ⁢ ( A ) {K_{n}(A)} , which is holomorphic symplectic but not simply connected, contrary to the case of K3 surfaces.

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

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 constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but only for polarizations with discriminant 1. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group of co-dimension two algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold Y of Kummer type onto the third intermediate Jacobian of Y, as predicted by the generalized Hodge Conjecture.

Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds

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 result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for Abelian varieties

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 address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let $\gamma$ be a correspondence between two abelian varieties $A$ and $B$ that can be written as a linear combination of products of symmetric divisors. Assume that $A$ is isogenous to the product of an abelian variety of totally real type with the power of an abelian surface. We show that $\gamma$ satisfies the conclusion of Bloch's conjecture. A key ingredient consists in establishing a strong form of the generalized Hodge conjecture for Hodge sub-structures of the cohomology of $A$ that arise as sub-representations of the Lefschetz group of $A$. As a by-product of our method, we use a strong form of the generalized Hodge conjecture established for powers of abelian surfaces to show that every finite-order symplectic automorphism of a generalized Kummer variety acts as the identity on the zero-cycles.

Bloch's conjecture for Enriques varieties

Enriques varieties have been defined as higher-dimensional generalizations of Enriques surfaces. Bloch's conjecture implies that Enriques varieties should have trivial Chow group of zero-cycles. We prove this is the case for all known examples of irreducible Enriques varieties of index larger than $2$. The proof is based on results concerning the Chow motive of generalized Kummer varieties.

Distinguished cycles on varieties with motive of abelian type and the Section Property

A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with a Chow motive of abelian type. We introduce the notion of a symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give a series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects.

Corrigendum and Addendum to “Polarized Parallel Transport and Uniruled Divisors on Generalized Kummer Varieties”

Abstract We correct the statement of the main result of [9] and provide some further precisions.

Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties

Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-G\"ottsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an algebra object in the category of Chow motives. Inspired by Ruan, one can formulate a motivic version of his Cohomological HyperK\"ahler Resolution Conjecture. We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura, in two situations related to an abelian surface $A$ and a positive integer $n$. Case (A) concerns Hilbert schemes of points of $A$ : the Chow motive of $A^{[n]}$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $[A^{n}/\mathfrak{S}_{n}]$. Case (B) for generalized Kummer varieties : the Chow motive of the generalized Kummer variety $K_n(A)$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $[A_{0}^{n+1}/\mathfrak {S}_{n+1}]$, where $A_{0}^{n+1}$ is the kernel abelian variety of the summation map $A^{n+1}\to A$. As a byproduct, we prove the original Cohomological HyperK\"ahler Resolution Conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow-K\"unneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch-Beilinson-Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville.

Action of correspondences on filtrations on cohomology and 0-cycles of Abelian varieties

We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which includes any abelian variety of dimension at most 5, powers of complex elliptic curves, etc.) that vanishes as a morphism on a certain quotient of its middle singular cohomology, then it vanishes as a morphism on the deepest part of a particular filtration on the Chow group of 0-cycles of the abelian variety. As a consequence, we prove that an automorphism of such an abelian variety that acts as the identity on a certain quotient of its middle singular cohomology acts as the identity on the deepest part of this filtration on the Chow group of 0-cycles of the abelian variety. As an application, we prove that for the generalized Kummer variety associated to a complex abelian surface and the automorphism induced from a symplectic automorphism of the complex abelian surface, the automorphism of the generalized Kummer variety acts as the identity on a certain subgroup of its Chow group of 0-cycles.

Polarized Parallel Transport and Uniruled Divisors on Deformations of Generalized Kummer Varieties

In this note we characterize polarized parallel transport operators on irreducible holomorphic symplectic varieties which are deformations of generalized Kummer varieties. We then apply such characterization to show the existence of ample uniruled divisors on these varieties and derive some interesting consequences on their Chow group of 0-cycles.

Algebraic Cycles on a Generalized Kummer Variety

We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying abelian surface. One application of this calculation is to show that the Hodge conjecture holds for arbitrary products of generalized Kummer varieties. As another application, all numerically trivial 1-cycles on arbitrary products of generalized Kummer varieties are smash-nipotent.

Hard Lefschetz for Chow groups of generalized Kummer varieties

The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen-Vial and Fu-Tian-Vial on multiplicative Chow-K\"unneth decompositions.

Tautological sheaves: Stability, moduli spaces and restrictions to generalised Kummer varieties

Results on stability of tautological sheaves on Hilbert schemes of points are extended to higher dimensions and to the restriction of tautological sheaves to generalised Kummer varieties. This provides a big class of new examples of stable sheaves on higher dimensional irreducible symplectic manifolds. Some aspects of deformations of tautological sheaves are studied.

On the Chow group of zero-cycles of a generalized Kummer variety

For a generalized Kummer variety X of dimension 2n, we will construct for each 0≤i≤n some co-isotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit filtration introduced by Voisin on the Chow group of zero-cycles of a generalized Kummer variety coincides with the induced Beauville decomposition from the Chow ring of abelian varieties. As a consequence, the rational orbit filtration is opposite to the conjectural Bloch–Beilinson filtration for generalized Kummer varieties.