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Delta-invariant for projective bundles over a curve and K-semistability

Houari Benammar Ammar, Louis Massonnet, Chenxi Yin

Abstract

Consider $E$ a vector bundle over a smooth curve $C$. We compute the $δ$-invariant of all ample ($\mathbb{Q}$-) line bundles on $\mathbb{P}(E)$ when $E$ is strictly Mumford semistable. We also investigate the case when one assumes that the Harder-Narasimhan filtration of $E$ has only one step.

Delta-invariant for projective bundles over a curve and K-semistability

Abstract

Consider a vector bundle over a smooth curve . We compute the -invariant of all ample (-) line bundles on when is strictly Mumford semistable. We also investigate the case when one assumes that the Harder-Narasimhan filtration of has only one step.

Paper Structure

This paper contains 9 sections, 23 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Basic setup
  3. A lower bound by Abban-Zhuang
  4. Zariski decomposition for projective bundles and $S(L,F)$
  5. Case $c=0$
  6. Case $c\neq 0$
  7. A combinatorial identity
  8. About $S(L,F)$ and $\delta(X,L)$
  9. Semistable vector bundles and K-semistability

Key Result

Theorem 1.1

Let $X$ be a klt Fano variety. The following hold:

Theorems & Definitions (41)

  • Theorem 1.1: BlumXupolydegeneration2019Fujitavaluative2019ChiLivolumemin2017
  • Theorem 1.2: Zhangquantization2024
  • Theorem 1.3: Zhangcontinuitydelta2021
  • Theorem 1.4: hattori2022k
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 2.1: effeccone,Miyaokaminimalvariety1987,nakayama2004zariski
  • Definition 2.2: fujita2018kstability
  • Remark 2.3
  • ...and 31 more