An approach towards the Koll\'ar-Peskine problem via the Instanton Moduli Space

TL;DR

This paper investigates a variant of Kollár and Peskine's problem on the triviality of vector bundles in families, linking it to the nonexistence of certain morphisms into infinite Grassmannians and Donaldson moduli spaces.

Contribution

It reduces the geometric problem to the nonexistence of specific equivariant morphisms into moduli spaces, connecting vector bundle triviality to algebraic geometry and gauge theory.

Findings

Equivalence between the triviality problem and morphism nonexistence.

Reduction to the nonexistence of $C^*$-equivariant maps into Donaldson moduli spaces.

Insight into the structure of vector bundles via infinite Grassmannian and moduli space mappings.

Abstract

We look at the following question raised by Koll\'ar and Peskine. (Actually, it is a slightly weaker version of their question.) Let $V_t$ be a family of rank two vector bundles on $\Bbb P^3$. Assume that the general member of the family is a trivial vector bundle. Then, is the special member $V_0$ also a trivial vector bundle? We show that this question is equivalent to the nonexistence of morphisms from $\Bbb P^3\to \mathcal{X}$, where $\mathcal{X}$ is the infinite Grassmannian associated to SL(2). We further reduce this question to the nonexistence of $\Bbb C^*$-equivariant morphisms from $\Bbb C^3\setminus \{0\} \to \mathcal{M}_d$ (for any $d>0$), where $\mathcal{M}_d$ is the Donaldson moduli space of isomorphism classes of rank two vector bundles $\mathcal{V}$ over $\Bbb P^2$ with trivial determinant and with second Chern class $d$ together with a trivialization of $\mathcal{V}_{|\Bbb P^1}$.