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Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, I: Construction of an invariant

Dai Imaike

Abstract

In this paper, we construct an invariant for irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type with antisymplectic involution by using the equivariant analytic torsion. Moreover, we give a formula for the complex Hessian of the logarithm of the invariant.

Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, I: Construction of an invariant

Abstract

In this paper, we construct an invariant for irreducible holomorphic symplectic manifolds of -type with antisymplectic involution by using the equivariant analytic torsion. Moreover, we give a formula for the complex Hessian of the logarithm of the invariant.

Paper Structure

This paper contains 13 sections, 33 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Analytic torsion and its fundamental properties
  3. Equivariant analytic torsion
  4. A fundamental property of equivariant analytic torsion
  5. Irreducible holomorphic symplectic manifolds and antisymplectic involutions
  6. Lattices
  7. Irreducible holomorphic symplectic manifolds and antisymplectic involutions
  8. Kähler-type chambers
  9. Relations of orthogonal modular varieties
  10. An invariant of $K3^{[2]}$-type manifolds with antisymplectic involution
  11. Some fundamental properties of hyperkähler manifolds
  12. Construction of an invariant
  13. Properties of $\tau_{M, \mathcal{K}}$

Key Result

Theorem 1

The real number $\tau_{M, \mathcal{K}}(X, \iota)$ is independent of the choice of an $\iota$-invariant Kähler form. In particular, $\tau_{M, \mathcal{K}}(X, \iota)$ is an invariant of $(X, \iota)$.

Theorems & Definitions (86)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 76 more