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

Guilfoyle, Brendan; Klingenberg, Wilhelm

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 565-576.

On etude l’espace des courbes holomorphes a bord dans une surface reelle situe dans une fibree vectoriel de rang 2 sur une variete reelle a dimension deux. On prouve que, si le fibree ambient admet une action transitive et holomorphe qui preserve la fibration, alors une section avec un et seulement un point complexe admet des deformation petits dans la norme $C^{2,\alpha }$ tel que toute disque holomorphe a bord dans la deformation soit Fredholm reguliere. On prouve aussi la Fredholm regularite dans le cas que le fibree ambient est Kaehlerien a signature (2,2), l’action de la groupe e holomorphe et symplectique, et la surface bordante est Lagrangienne avec un seul poit complexe.

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 Kähler, the action is both holomorphic and symplectic, and the section is Lagrangian with a single complex point.

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