Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homogeneous Ulrich bundles on isotropic flag varieties

Xinyi Fang, Yusuke Nakayama

Abstract

In this paper, we consider the existence problem of Ulrich bundles on a rational homogeneous space $G/P$ of type $B$, $C$ or $D$. We show that if the Picard number of $G/P$ is greater than or equal to $2$, then there are no irreducible homogeneous Ulrich bundles on $G/P$ with respect to the minimal ample class.

Homogeneous Ulrich bundles on isotropic flag varieties

Abstract

In this paper, we consider the existence problem of Ulrich bundles on a rational homogeneous space of type , or . We show that if the Picard number of is greater than or equal to , then there are no irreducible homogeneous Ulrich bundles on with respect to the minimal ample class.

Paper Structure

This paper contains 15 sections, 20 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Notations and conventions
  4. Preliminaries
  5. Rational homogeneous spaces
  6. Homogeneous vector bundles
  7. Criterion for irreducible homogeneous bundles to be Ulrich
  8. Homogeneous Ulrich bundles on isotropic flag varieties
  9. Type $B_n$ or $C_n$
  10. $J=\{d_1,\ldots,d_s\}$ with $d_s\ne n$
  11. $J=\{d_1,\ldots,d_s\}$ with $d_s=n$
  12. Type $D_n$
  13. $J=\{d_1,\ldots,d_s\}$ with $d_s\le n-2$
  14. $J=\{d_1,\ldots,d_s\}$ with $d_s=n-1$ or $d_s=n,~d_{s-1}\ne n-1$
  15. $J=\{d_1,\ldots,d_s\}$ with $d_{s-1}=n-1$ and $d_s=n$

Key Result

Theorem 1.1

Let $G/P$ be an isotropic flag variety. If the Picard number of $G/P$ is greater than or equal to $2$, then there are no irreducible homogeneous Ulrich bundles on $G/P$ with respect to the minimal ample class.

Theorems & Definitions (38)

  • Theorem 1.1: Theorem \ref{['mainthm']}
  • Definition 2.1
  • Lemma 2.2: Ott Theorem $9.7$
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5: Fang Theorem $1.3$,Nakayama
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 28 more