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Pullback and Direct Image of Parabolic Ample and Parabolic Nef Vector Bundles

Ashima Bansal, Indranil Biswas

TL;DR

This work analyzes how positivity properties of parabolic vector bundles on irreducible smooth projective curves behave under morphisms. It develops criteria for parabolic ampleness and nefness via Harder–Narasimhan data and parabolic degrees, and then proves that under a non-constant morphism, the parabolic positivity is preserved by pullback and pushforward: $E_\nast$ is parabolic ample (nef) iff $f^{*}E_\nast$ is, and similarly $f_{*}E_\nast$ preserves these properties. The results extend to parabolic anti-versions through duality, with the pullback/duality interplay and a Galois-closure approach for direct images. Collectively, the paper provides a coherent framework for the behavior of parabolic positivity under morphisms, enabling robust construction and manipulation of parabolic bundles on curves. The findings have implications for the study of parabolic stability, positivity, and birational geometry in the parabolic setting.

Abstract

We prove that under a finite surjective map of irreducible smooth complex projective curves, the pullback and direct image of a parabolic ample (respectively, parabolic nef) vector bundle is again parabolic ample (respectively, parabolic nef) if and only if the original parabolic vector bundle is parabolic ample (respectively, parabolic nef).

Pullback and Direct Image of Parabolic Ample and Parabolic Nef Vector Bundles

TL;DR

This work analyzes how positivity properties of parabolic vector bundles on irreducible smooth projective curves behave under morphisms. It develops criteria for parabolic ampleness and nefness via Harder–Narasimhan data and parabolic degrees, and then proves that under a non-constant morphism, the parabolic positivity is preserved by pullback and pushforward: is parabolic ample (nef) iff is, and similarly preserves these properties. The results extend to parabolic anti-versions through duality, with the pullback/duality interplay and a Galois-closure approach for direct images. Collectively, the paper provides a coherent framework for the behavior of parabolic positivity under morphisms, enabling robust construction and manipulation of parabolic bundles on curves. The findings have implications for the study of parabolic stability, positivity, and birational geometry in the parabolic setting.

Abstract

We prove that under a finite surjective map of irreducible smooth complex projective curves, the pullback and direct image of a parabolic ample (respectively, parabolic nef) vector bundle is again parabolic ample (respectively, parabolic nef) if and only if the original parabolic vector bundle is parabolic ample (respectively, parabolic nef).
Paper Structure (4 sections, 9 theorems, 49 equations)

This paper contains 4 sections, 9 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Criteria for parabolic ample and Parabolic nef vector bundles
  3. Pullback of parabolic ample and parabolic nef vector bundles
  4. Direct image of parabolic ample and parabolic nef vector bundles

Key Result

Lemma 2.1

Let $E_{\ast}$ be a parabolic vector bundle on $X$ with the Harder-Narasimhan filtration as in hn. Let $Q_{\ast}$ be any quotient parabolic vector bundle of $E_{\ast}$. Then $\emph{par-}\mu(Q_{\ast})\,\geq\,$par-${\mu}_{\emph{min}}(E_{\ast})$.

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 12 more