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K3 surfaces associated with varieties of generalized Kummer type

Salvatore Floccari

TL;DR

The paper establishes a geometric bridge between varieties of generalized Kummer type and associated K3 surfaces S_K by constructing a W_K of K3^[m]-type inside Kum^n-varieties with a primitive, doubled pull-back embedding into H^2. This yields an algebraic Kuga–Satake correspondence for S_K and, via existing work of O'Grady, Markman, Voisin and Varesco, proves the Hodge conjecture for all powers of S_K and for abelian fourfolds of Weil type with discriminant 1 and their powers. The approach leverages motivic formalism, deformation theory, moduli spaces of stable sheaves on K3 surfaces, and functorial properties of the Kuga–Satake construction to transfer algebraicity from KS(S_K) to S_K and to related Weil-type abelian varieties. Consequently, a broad class of K3 surfaces associated to Kum^n-type varieties satisfy the Hodge conjecture for their powers, with explicit isogeny relations to Weil fourfolds enhancing understanding of Hodge classes in this landscape.

Abstract

With any hyper-Kähler 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 fundamentally on the works of O'Grady, Markman, Voisin, Varesco, we establish the Hodge conjecture for all powers of any of these K3 surfaces as well as for all abelian fourfolds of Weil type with discriminant 1 and their powers, strenghtening a result of Markman.

K3 surfaces associated with varieties of generalized Kummer type

TL;DR

The paper establishes a geometric bridge between varieties of generalized Kummer type and associated K3 surfaces S_K by constructing a W_K of K3^[m]-type inside Kum^n-varieties with a primitive, doubled pull-back embedding into H^2. This yields an algebraic Kuga–Satake correspondence for S_K and, via existing work of O'Grady, Markman, Voisin and Varesco, proves the Hodge conjecture for all powers of S_K and for abelian fourfolds of Weil type with discriminant 1 and their powers. The approach leverages motivic formalism, deformation theory, moduli spaces of stable sheaves on K3 surfaces, and functorial properties of the Kuga–Satake construction to transfer algebraicity from KS(S_K) to S_K and to related Weil-type abelian varieties. Consequently, a broad class of K3 surfaces associated to Kum^n-type varieties satisfy the Hodge conjecture for their powers, with explicit isogeny relations to Weil fourfolds enhancing understanding of Hodge classes in this landscape.

Abstract

With any hyper-Kähler variety of generalized Kummer type is associated via Hodge theory a K3 surface . We show how they are related geometrically through a moduli space of sheaves on . As a consequence, building fundamentally on the works of O'Grady, Markman, Voisin, Varesco, we establish the Hodge conjecture for all powers of any of these K3 surfaces as well as for all abelian fourfolds of Weil type with discriminant 1 and their powers, strenghtening a result of Markman.
Paper Structure (5 sections, 22 theorems, 28 equations)

This paper contains 5 sections, 22 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Motives
  3. The Kuga-Satake construction
  4. Some $\mathrm{K}3^{[m]}$-type submanifolds of generalized Kummer varieties
  5. K3 surfaces associated with varieties of generalized Kummer type

Key Result

Theorem 1.1

There exists an algebraic cycle on $S_K\times K$ inducing a Hodge isometry

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Theorem 2.2: Ara06
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: foster
  • Remark 3.1
  • ...and 45 more