An approach towards the Kollár-Peskine problem via the Instanton Moduli Space

Abstract

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

12 pages