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Counterexamples to the Kuznetsov--Shinder L-equivalence conjecture

Reinder Meinsma

TL;DR

The paper refutes the Kuznetsov–Shinder conjecture by constructing hyperkähler counterexamples that are derived-equivalent but not $L$-equivalent, via moduli spaces of sheaves on K3 surfaces. It develops lattice-based invariants—the gluing group $G(X)$ and coarseness $ ext{crs}(X)$—that are preserved under $L$-equivalence and shows they obstruct $L$-equivalence in the presented moduli spaces. The key finding is that for certain $K3^{[n]}$-type manifolds, $|G(X)|$ is invariant under $L$-equivalence while the divisibility data of Mukai vectors can differ, yielding $D$-equivalence without $L$-equivalence. This provides a concrete negative answer to the conjecture in the higher-dimensional hyperkähler setting and clarifies the role of transcendental lattice data and Brauer obstructions in $L$-equivalence. The results illuminate the nuanced relationship between derived categories and motivic invariants in complex geometry.

Abstract

We disprove a conjecture of Kuznetsov--Shinder, which posits that $D$-equivalent simply connected varieties are $L$-equivalent, by constructing a counterexample using moduli spaces of sheaves on K3 surfaces.

Counterexamples to the Kuznetsov--Shinder L-equivalence conjecture

TL;DR

The paper refutes the Kuznetsov–Shinder conjecture by constructing hyperkähler counterexamples that are derived-equivalent but not -equivalent, via moduli spaces of sheaves on K3 surfaces. It develops lattice-based invariants—the gluing group and coarseness —that are preserved under -equivalence and shows they obstruct -equivalence in the presented moduli spaces. The key finding is that for certain -type manifolds, is invariant under -equivalence while the divisibility data of Mukai vectors can differ, yielding -equivalence without -equivalence. This provides a concrete negative answer to the conjecture in the higher-dimensional hyperkähler setting and clarifies the role of transcendental lattice data and Brauer obstructions in -equivalence. The results illuminate the nuanced relationship between derived categories and motivic invariants in complex geometry.

Abstract

We disprove a conjecture of Kuznetsov--Shinder, which posits that -equivalent simply connected varieties are -equivalent, by constructing a counterexample using moduli spaces of sheaves on K3 surfaces.

Paper Structure

This paper contains 6 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The Grothendieck group of Hodge structures
  3. Transcendental Hodge substructures and gluing groups
  4. Hodge lattices of K3-type
  5. The coarseness of a hyperkähler manifold
  6. Counterexamples to the Kuznetsov--Shinder conjecture

Key Result

Theorem 1.3

Let $(S,H)$ be a primitively polarised complex K3 surface of degree $H^2 = 2g-2 \geq 2$, and assume that $S$ has Picard rank 1. Then the moduli spaces of stable sheaves $\mathop{\mathrm{\overline{Pic}}}\nolimits^0\coloneqq M(0,H,1-g)$ and $\mathop{\mathrm{\overline{Pic}}}\nolimits^{g-1} = M(0,H,0)$

Theorems & Definitions (26)

  • Definition 1.1: KS18
  • Conjecture 1.2
  • Theorem 1.3: See Theorem \ref{['thm: D equivalence does not imply L equivalence Main Text']}
  • Theorem 1.4: See Theorem \ref{['thm: L equivalent same divisibility']} and Corollary \ref{['cor: L equivalent moduli spaces have same divisibility']}
  • Conjecture 1.5
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 16 more