Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Direct image of structure sheaf and parabolic stability

Indranil Biswas, Manish Kumar, A. J. Parameswaran

Abstract

Let $f : X \rightarrow Y$ be a dominant generically smooth morphism between irreducible smooth projective curves over an algebraically closed field $k$ such that ${\rm Char}(k)> \text{degree}(f)$ if the characteristic of $k$ is nonzero. We prove that $(f_*{\mathcal O}_X)/{\mathcal O}_Y$ equipped with a natural parabolic structure is parabolic polystable. Several conditions are given that ensure that the parabolic vector bundle $(f_*{\mathcal O}_X)/{\mathcal O}_Y$ is actually parabolic stable.

Direct image of structure sheaf and parabolic stability

Abstract

Let be a dominant generically smooth morphism between irreducible smooth projective curves over an algebraically closed field such that if the characteristic of is nonzero. We prove that equipped with a natural parabolic structure is parabolic polystable. Several conditions are given that ensure that the parabolic vector bundle is actually parabolic stable.

Paper Structure

This paper contains 9 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Direct image and parabolic structure
  3. Direct image of structure sheaf
  4. Parabolic bundles
  5. Parabolic structure on a direct image
  6. Parabolic pullback and polystability of direct image
  7. Pullback of parabolic bundles
  8. Polystability of direct image
  9. Parabolic stability

Key Result

Lemma 3.1

The parabolic degree of the parabolic vector bundle $(f_*{\mathcal{O}}_X)_*$ is zero.

Theorems & Definitions (14)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 4 more