Castelnuovo's bound and rigidity in almost complex geometry

TL;DR

This paper investigates whether energy bounds can imply genus bounds for pseudo-holomorphic curves in almost complex manifolds, establishing new results in 6-dimensional symplectic Calabi-Yau manifolds using compactness and regularity theorems.

Contribution

It provides the first genus bound result in dimension 6 for symplectic Calabi-Yau manifolds, extending known results from other dimensions.

Findings

Energy bounds can imply genus bounds in dimension 6

New compactness and regularity theorems for J-holomorphic currents

Establishment of genus bounds in symplectic Calabi-Yau 6-manifolds

Abstract

This article is concerned with the question of whether an energy bound implies a genus bound for pseudo-holomorphic curves in almost complex manifolds. After reviewing what is known in dimensions other than 6, we establish a new result in this direction in dimension 6; in particular, for symplectic Calabi-Yau 6-manifolds. The proof relies on compactness and regularity theorems for J-holomorphic currents.