H-instanton bundles on three-dimensional polarized projective varieties

TL;DR

This paper introduces a generalized concept of instanton bundles on three-dimensional polarized projective varieties, providing constructions, stability results, and moduli space descriptions for these bundles on various classes of threefolds.

Contribution

It defines $H$-instanton bundles on polarized threefolds, extends their construction to rational normal scrolls, and analyzes their stability and moduli spaces.

Findings

Existence of $ ext{μ}$-stable $H$-instanton bundles on rational normal scrolls.

Monadic descriptions of $H$-instanton bundles.

Identification of a component of the moduli space of $ ext{μ}$-stable bundles.

Abstract

We propose a notion of instanton bundle (called $H$-instanton bundle) on any projective variety of dimension three polarized by a very ample divisor $H$, that naturally generalizes the ones on $\mathbb{P}^3$ and on the flag threefold $F(0,1,2)$. We discuss the cases of Veronese and Fano threefolds. Then we deal with $H$-instanton bundles $\mathcal{E}$ on three-dimensional rational normal scrolls $S(a_0,a_1,a_2)$. We give a monadic description of $H$-instanton bundles and we prove the existence of $\mu$-stable $H$-instanton bundles on $S(a_0,a_1,a_2)$ for any admissible charge $k=c_2(\mathcal{E})H$. Then we deal in more detail with $S(a,a,b)$ and $S(a_0,a_1,a_2)$ with $a_0+a_1>a_2$ and even degree. Finally we describe a nice component of the moduli space of $\mu$-stable bundles whose points represent $H$-instantons.