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On the triviality of direct image of vector bundles

Indranil Biswas, Jagadish Pine

TL;DR

The paper investigates when a finite morphism $\pi: X \to Y$ admits a vector bundle $E$ on $X$ with trivial $\pi_*E$, introducing the notions of relatively bundles and relatively Ulrich bundles. It establishes that the branching data of $\pi$ governs existence, and proves a complete criterion for ramified abelian Galois covers, including an application showing ramified abelian Galois coverings of $\mathbb{P}^n$ support Ulrich bundles. A key contribution is both necessary and sufficient criteria linked to the branch divisors via $d$-fold multiplication maps and global generation of associated line bundles $M_i$, with constructive proofs using matrix factorizations for cyclic and double covers. The results yield new Ulrich-bundle examples on ramified abelian Galois coverings of projective space and illuminate connections to the Eisenbud–Schreyer framework on the existence of vector bundles.

Abstract

Let $π\,:\, X \,\longrightarrow\, Y$ be a finite morphism of smooth projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $π$ such that there exists a vector bundle $E$ on $X$ whose direct image $π_*E$ is trivial. We show that the existence of $E$ is guided by the properties of the branching divisor of $π$. When the covering $π\,:\, X \,\longrightarrow\, Y$ is ramified abelian Galois, we give a complete answer. As an application, we prove every smooth ramified abelian Galois covering of $\mathbb{P}^n$ supports an Ulrich bundle.

On the triviality of direct image of vector bundles

TL;DR

The paper investigates when a finite morphism admits a vector bundle on with trivial , introducing the notions of relatively bundles and relatively Ulrich bundles. It establishes that the branching data of governs existence, and proves a complete criterion for ramified abelian Galois covers, including an application showing ramified abelian Galois coverings of support Ulrich bundles. A key contribution is both necessary and sufficient criteria linked to the branch divisors via -fold multiplication maps and global generation of associated line bundles , with constructive proofs using matrix factorizations for cyclic and double covers. The results yield new Ulrich-bundle examples on ramified abelian Galois coverings of projective space and illuminate connections to the Eisenbud–Schreyer framework on the existence of vector bundles.

Abstract

Let be a finite morphism of smooth projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for such that there exists a vector bundle on whose direct image is trivial. We show that the existence of is guided by the properties of the branching divisor of . When the covering is ramified abelian Galois, we give a complete answer. As an application, we prove every smooth ramified abelian Galois covering of supports an Ulrich bundle.
Paper Structure (4 sections, 16 theorems, 55 equations)

This paper contains 4 sections, 16 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Examples of relatively bundles
  3. Necessary conditions for relatively bundles
  4. Sufficient criterion for relatively Ulrich bundles

Key Result

Theorem 1.3

There exists a relatively Ulrich bundle $E$ on $X$ with respect to $\pi$ if and only if the branch divisor $B_i \,\in\, H^0(Y,\, M_i^{\otimes d_i})$ is in the image of the $d_i$-fold multiplication map for all $1 \,\le\, i \,\le\, l$.

Theorems & Definitions (33)

  • Definition 1.2: MOHANKUMAR2025107946
  • Theorem 1.3: Proposition \ref{['dfoldsurj']} and Theorem \ref{['sufficient']}
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Example 2.5
  • ...and 23 more