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The Lefschetz standard conjectures for 4d-dimensional Kummer-type hyper-Kaehler varieties

Josiah Foster

TL;DR

The paper proves the Lefschetz standard conjecture for 4d-dimensional IHSMs of generalized Kummer deformation type under the Buchweitz–Flenner conjecture, by constructing and deforming semiregular, inflated semihomogeneous bundles and their Fourier–Mukai kernels to yield derived equivalences between Kum-type IHSMs. It connects these equivalences to the BKR correspondence to transport algebraic Lefschetz inverses across deformation spaces, and employs the LLV algebra and Taelman’s framework to ensure degree-reversing Hodge isometries induce algebraic Lefschetz operators. The work also develops a program of deforming kernels along moduli of rational Hodge isometries and using twistor theory to control the Hodge-type of characteristic classes κ across families, paving the way for broader LSC results in Kummer-type IHSMs. Finally, the paper outlines potential future avenues, including new derived autoequivalences and higher-dimensional cases (e.g., dimension 10) that could yield further progress on the standard conjectures for Kummer-type IHSMs.

Abstract

Contingent upon a conjecture of Buchweitz and Flenner, we prove the Lefschetz standard conjectures for 4d-dimensional projective varieties of generalized Kummer deformation type.

The Lefschetz standard conjectures for 4d-dimensional Kummer-type hyper-Kaehler varieties

TL;DR

The paper proves the Lefschetz standard conjecture for 4d-dimensional IHSMs of generalized Kummer deformation type under the Buchweitz–Flenner conjecture, by constructing and deforming semiregular, inflated semihomogeneous bundles and their Fourier–Mukai kernels to yield derived equivalences between Kum-type IHSMs. It connects these equivalences to the BKR correspondence to transport algebraic Lefschetz inverses across deformation spaces, and employs the LLV algebra and Taelman’s framework to ensure degree-reversing Hodge isometries induce algebraic Lefschetz operators. The work also develops a program of deforming kernels along moduli of rational Hodge isometries and using twistor theory to control the Hodge-type of characteristic classes κ across families, paving the way for broader LSC results in Kummer-type IHSMs. Finally, the paper outlines potential future avenues, including new derived autoequivalences and higher-dimensional cases (e.g., dimension 10) that could yield further progress on the standard conjectures for Kummer-type IHSMs.

Abstract

Contingent upon a conjecture of Buchweitz and Flenner, we prove the Lefschetz standard conjectures for 4d-dimensional projective varieties of generalized Kummer deformation type.

Paper Structure

This paper contains 22 sections, 44 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. The standard conjectures
  3. Kummer-type irreducible holomorphic symplectic manifolds
  4. The main results
  5. Outline of the paper
  6. Semiregularity
  7. Buchweitz-Flenner theorems
  8. Twisted sheaves and semiregularity
  9. Simple semihomogeneous vector bundles on products of Abelian varieties
  10. Semiregular vector bundles on products of Kummer-type IHSMs
  11. Equivariant derived categories
  12. Inflation
  13. The BKR correspondence
  14. Derived equivalences of Kummer-type IHSMs
  15. Semiregularity on products
  16. ...and 7 more sections

Key Result

Theorem 1.6

Let $A$ be an Abelian surface an let $v\in H^{ev}(A, \mathbb{Z})$ be a primitive Mukai vector that represents a sheaf. Assume $v^2\geq 6$. Then a fiber over $a\in A\times \widehat{A}$ of the Albanese morphism is a variety $Kum_a(v)$ of generalized Kummer deformation type.

Theorems & Definitions (112)

  • Conjecture 1.1: Gr
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6: YoshiokaAbelian
  • Theorem 1.7
  • Remark 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 102 more