Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Eunjeong Lee, Kyeong-Dong Park

Abstract

We classify fourfolds with trivial canonical bundle which are zero loci of general global sections of completely reducible equivariant vector bundles over exceptional homogeneous varieties of Picard number one. By computing their Hodge numbers, we see that there exist no hyperkähler fourfolds among them. This implies that a hyperkähler fourfold represented as the zero locus of a general global section of a completely reducible equivariant vector bundle over a rational homogeneous variety of Picard number one is one of the two cases described by Beauville--Donagi and Debarre--Voisin.

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Abstract

We classify fourfolds with trivial canonical bundle which are zero loci of general global sections of completely reducible equivariant vector bundles over exceptional homogeneous varieties of Picard number one. By computing their Hodge numbers, we see that there exist no hyperkähler fourfolds among them. This implies that a hyperkähler fourfold represented as the zero locus of a general global section of a completely reducible equivariant vector bundle over a rational homogeneous variety of Picard number one is one of the two cases described by Beauville--Donagi and Debarre--Voisin.

Paper Structure

This paper contains 9 sections, 22 theorems, 71 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Hyperkähler manifolds and equivariant vector bundles
  3. Hyperkähler manifolds and irreducible holomorphic symplectic manifolds
  4. Equivariant vector bundles over homogeneous varieties
  5. Classification of fourfolds with trivial canonical bundle
  6. Computations of Hodge numbers
  7. Computations of $h^{0, q}(Z)$
  8. Computations of $h^{1, q}(Z)$
  9. Computations of $h^{2, 2}(Z)$

Key Result

Theorem 1.1

Let $G/P$ be a rational homogeneous variety of Picard number one, where $G$ is a complex simple Lie group of exceptional type and $P \subset G$ is a maximal parabolic subgroup. For a completely reducible, globally generated, equivariant vector bundle $\mathcal{F}$ over $G/P$, if $Z \subset G/P$ is a

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5: Beauville83
  • Corollary 2.6
  • ...and 44 more