# Fredholm-Regularity of Holomorphic Discs in Plane Bundles over Compact Surfaces

## TL;DR

This paper investigates conditions under which holomorphic discs in certain 4-manifolds are Fredholm regular, focusing on cases with specific symmetries and geometric structures, enhancing understanding of their moduli spaces.

## Contribution

It establishes Fredholm regularity for holomorphic discs in plane bundles over compact surfaces under symmetry and geometric conditions, extending previous regularity results.

## Key findings

- Fredholm regularity holds for discs with boundary on sections with a single complex point.
- Regularity is proven in the presence of a fibre-preserving transitive holomorphic action.
- Results include cases with neutral Kähler structures and Lagrangian sections.

## Abstract

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has $C^{2,\alpha}$-close sections such that any (non-multiply covered) holomorphic disc with boundary in these sections are Fredholm regular. Fredholm regularity is also established when the complex surface is neutral K\"ahler, the action is both holomorphic and symplectic, and the section is Lagrangian with a single complex point.