New results in the analysis of pseudo-holomorphic curves

Abstract

In this thesis, we present new results on three aspects of moduli spaces of pseudoholomorphic curves: smoothness, compactness and bifurcations.In the first part of this thesis, dealing with smoothness, we give a functorial construction of a so-called relative smooth structure on the moduli spaces of solutions to the (perturbed) pseudo-holomorphic curve equation. In the second part of this thesis, dealing with compactness, we prove a quantitative version of Gromov’s compactness theorem for closed pseudo-holomorphic curves of genus 0 in a symplectic manifold. In the third part of this thesis, dealing with bifurcations, we study moduli spaces of embedded pseudo-holomorphic curves in a Calabi–Yau 3-fold. Performing a careful bifurcation analysis of these moduli spaces in generic 1-parameter families leads, in some cases, to the construction of an integer valued invariant of Calabi–Yau 3-folds which counts embedded curves with suitably defined integer weights. This part is joint with Shaoyun Bai.