Ample divisor complements, Floer spectra, and relative Gromov-Witten theory

TL;DR

This paper develops a spectral lift of a morphism to analyze Floer homotopy types of ample divisor complements, revealing obstructions linked to Gromov-Witten invariants and providing explicit examples for hypersurfaces.

Contribution

It introduces a spectral lift of the log PSS morphism and connects splitting obstructions to stable homotopy classes via relative Gromov-Witten theory, with explicit computations.

Findings

Computed the associated graded Floer homotopy types for ample divisor complements.

Identified obstructions to splitting as stable homotopy classes from Gromov-Witten moduli spaces.

Provided examples including affine parts of smooth projective hypersurfaces.

Abstract

We spectrally lift Ganatra-Pomerleano's low-energy log PSS morphism to compute the associated graded of Floer homotopy types of ample smooth divisor complements. Moreover, we show the obstruction to splitting into the associated graded is encoded in a stable homotopy class defined via (higher-dimensional) genus 0 relative Gromov-Witten moduli spaces. We compute numerous examples of splittings, including the affine part of all smooth projective hypersurfaces of degree at least 2.