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On numerically trivial automorphisms of compact hyperkähler manifolds of dimension 4

Chen Jiang, Wenfei Liu

TL;DR

The paper proves that a compact hyperkähler manifold of complex dimension $4$ has a faithful cohomological action of its automorphism group, i.e. $\operatorname{Aut}_{\mathbb{Q}}(X)$ is trivial. The authors analyze quotients by symplectic automorphisms, describe fixed loci in dimension $4$, and construct a crepant partial resolution $W$ whose Betti numbers relate to the quotient via explicit formulas. They then establish the existence of line bundles with vanishing cohomology on such resolutions, which, combined with Nieper's Riemann–Roch computations, imposes strong topological restrictions. These ingredients yield a contradiction unless $\operatorname{Aut}_{\mathbb{Q}}(X)$ is trivial; the paper also discusses automorphisms acting trivially on integral cohomology and provides singular examples where such automorphisms exist.

Abstract

We prove that the automorphism group of a compact hyperkähler manifold of dimension 4 acts faithfully on the cohomology ring.

On numerically trivial automorphisms of compact hyperkähler manifolds of dimension 4

TL;DR

The paper proves that a compact hyperkähler manifold of complex dimension has a faithful cohomological action of its automorphism group, i.e. is trivial. The authors analyze quotients by symplectic automorphisms, describe fixed loci in dimension , and construct a crepant partial resolution whose Betti numbers relate to the quotient via explicit formulas. They then establish the existence of line bundles with vanishing cohomology on such resolutions, which, combined with Nieper's Riemann–Roch computations, imposes strong topological restrictions. These ingredients yield a contradiction unless is trivial; the paper also discusses automorphisms acting trivially on integral cohomology and provides singular examples where such automorphisms exist.

Abstract

We prove that the automorphism group of a compact hyperkähler manifold of dimension 4 acts faithfully on the cohomology ring.
Paper Structure (7 sections, 9 theorems, 19 equations)

This paper contains 7 sections, 9 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symplectic varieties
  4. Automorphisms acting trivially on certain cohomology groups
  5. Quotients of compact hyperkähler manifolds by symplectic automrophisms
  6. Existence of line bundles with vanishing cohomology
  7. Triviality of numerically trivial automorphism group

Key Result

Theorem 1.1

Let $X$ be a compact hyperkähler manifold of dimension $4$. Then $\operatorname{Aut}_{\mathbb Q}(X)$ is trivial. In other words, $\operatorname{Aut}(X)$ acts faithfully on $H^*(X, \mathbb{Q})$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Example 2.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 13 more