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Applied Mathematics | Physical Sciences and Mathematics
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 C2,α-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.
Brendan Guilfoyle; Wilhelm Klingenberg. Fredholm-Regularity of Holomorphic Discs in Plane Bundles over Compact Surfaces. Annals of the Faculty of Sciences of Toulouse: Mathematics, Series 6, Tome 29 (2020) no. 3, pp. 565-576. doi: 10.5802 / afst.1639.