Stable Wild Vafa-Witten Bundles on the Projective Plane

TL;DR

This paper studies the geometry and moduli space of stable wild Vafa-Witten bundles on the complex projective plane, including classification and fixed point analysis under a natural group action.

Contribution

It computes the dimension of the moduli space of stable pairs and classifies certain stable pairs with specific bundle structures.

Findings

Computed the dimension of the moduli space of stable pairs.

Classified stable pairs where the bundle splits or is a push-forward.

Analyzed the fixed point locus of the $ ext{C}^*$-action on the moduli space.

Abstract

This work explores the geometry of stable wild Vafa-Witten bundles over the complex projective plane $\mathbb{P}^2$. Specifically, we consider stable rank-two pairs $(E,\Phi)$, with $E\to\mathbb{P}^2$ a rank-two holomorphic vector bundle and $\Phi\in H^0(\mathbb{P}^2,\mathrm{End}_0E\otimes\mathcal{O}(d))$ for $d\geq0$, and compute the dimension of the moduli space of such stable pairs. Moreover, we classify stable pairs $(E,\Phi)$ when the underlying rank-two bundle $E$ splits or is the push-forward of a line bundle on $\mathbb{P}^1\times\mathbb{P}^1$. Lastly, we examine the fixed point locus of the natural $\mathbb{C}^*$-action on the moduli space.