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

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 $\mathcal{V}_{|\Bbb P^1}$.