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

Local Lagrangian Floer Homology of Quasi-Minimally Degenerate Intersections