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On L-equivalence for K3 surfaces and hyperkähler manifolds

Reinder Meinsma

TL;DR

The paper investigates when $L$-equivalence implies $D$-equivalence for K3 surfaces and hyperkähler manifolds, leveraging Efimov's Hodge-theoretic framework and the Derived Torelli Theorem. It introduces a mechanism to twist irreducible Hodge lattices by rational endomorphisms, linking Hodge-structure isomorphisms to Hodge isometries, and uses this to show that very general $L$-equivalent K3 surfaces are $D$-equivalent under suitable endomorphism constraints. The authors establish a precise criterion: if $X$ is a K3 surface with $ ho eq 18$ and $ ext{End}(T(X)_Q) frac{ig}{=}Q$, then $L$-equivalence with $Y$ forces $D$-equivalence; they also obtain partial extensions to hyperkähler fourfolds of K3$^{[n]}$-type and to moduli spaces of sheaves on K3 surfaces, highlighting the conditions under which $L$-equivalence implies (twisted) derived equivalence or birational relations. Overall, the work clarifies when the two notions align and when counterexamples arise, advancing the understanding of derived and motivic relationships in higher-dimensional geometry.

Abstract

This paper explores the relationship between L-equivalence and D-equivalence for K3 surfaces and hyperkähler manifolds. Building on Efimov's approach using Hodge theory, we prove that very general L-equivalent K3 surfaces are D-equivalent, leveraging the Derived Torelli Theorem for K3 surfaces. Our main technical contribution is that two distinct lattice structures on an integral, irreducible Hodge structure are related by a rational endomorphism of the Hodge structure. We partially extend our results to hyperkähler fourfolds and moduli spaces of sheaves on K3 surfaces.

On L-equivalence for K3 surfaces and hyperkähler manifolds

TL;DR

The paper investigates when -equivalence implies -equivalence for K3 surfaces and hyperkähler manifolds, leveraging Efimov's Hodge-theoretic framework and the Derived Torelli Theorem. It introduces a mechanism to twist irreducible Hodge lattices by rational endomorphisms, linking Hodge-structure isomorphisms to Hodge isometries, and uses this to show that very general -equivalent K3 surfaces are -equivalent under suitable endomorphism constraints. The authors establish a precise criterion: if is a K3 surface with and , then -equivalence with forces -equivalence; they also obtain partial extensions to hyperkähler fourfolds of K3-type and to moduli spaces of sheaves on K3 surfaces, highlighting the conditions under which -equivalence implies (twisted) derived equivalence or birational relations. Overall, the work clarifies when the two notions align and when counterexamples arise, advancing the understanding of derived and motivic relationships in higher-dimensional geometry.

Abstract

This paper explores the relationship between L-equivalence and D-equivalence for K3 surfaces and hyperkähler manifolds. Building on Efimov's approach using Hodge theory, we prove that very general L-equivalent K3 surfaces are D-equivalent, leveraging the Derived Torelli Theorem for K3 surfaces. Our main technical contribution is that two distinct lattice structures on an integral, irreducible Hodge structure are related by a rational endomorphism of the Hodge structure. We partially extend our results to hyperkähler fourfolds and moduli spaces of sheaves on K3 surfaces.
Paper Structure (7 sections, 15 theorems, 32 equations)

This paper contains 7 sections, 15 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Lattices and Hodge structures
  3. Lattices
  4. K3 surfaces and hyperkähler manifolds
  5. Irreducible Hodge lattices
  6. Hodge isomorphisms and Hodge isometries
  7. L-equivalence and D-equivalence

Key Result

Proposition 1.1

Let $T_1$ and $T_2$ be two irreducible Hodge lattices of K3 type. If $T_1$ and $T_2$ are isomorphic as Hodge structures, then there exists a rational Hodge automorphism $\phi\colon (T_1)_\mathbb{Q}\simeq (T_1)_\mathbb{Q}$ with the property that $T_2$ is Hodge isometric to $(T_1)_\phi$, which is the

Theorems & Definitions (34)

  • Proposition 1.1: see Proposition \ref{['prop: F H bijection']} and Lemma \ref{['lem: isomorphic hodge structure means twisted by endomorphism']}
  • Theorem 1.2: See Theorem \ref{['thm: L equivalent implies D equivalent in general']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 24 more