Local Lagrangian Floer Homology of Quasi-Minimally Degenerate Intersections

TL;DR

This paper introduces a new class of local Lagrangian intersections called quasi-minimally degenerate (QMD), develops methods to analyze their Floer homology, and constructs a spectral sequence linking local data to global Floer homology.

Contribution

The paper defines QMD intersections, develops local Floer homology techniques for them, and constructs a spectral sequence to compute Floer homology from local QMD data, extending previous frameworks.

Findings

Spectral sequence converges to Floer homology for QMD intersections.

Local Floer homology can be computed from singular homologies of boundary submanifolds.

Applications include new insights into affine varieties.

Abstract

We define a broad class of local Lagrangian intersections which we call quasi-minimally degenerate (QMD) before developing techniques for studying their local Floer homology. In some cases, one may think of such intersections as modeled on minimally degenerate functions as defined by Kirwan. One major result of this paper is: if $L_0,L_1$ are two Lagrangian submanifolds whose intersection decomposes into QMD sets, there is a spectral sequence converging to their Floer homology $HF_*(L_0,L_1)$ whose $E^1$ page is obtained from local data given by the QMD pieces. The $E^1$ terms are the singular homologies of submanifolds with boundary that come from perturbations of the QMD sets. We then give some applications of these techniques towards studying affine varieties, reproducing some prior results using our more general framework.