Overview
The research investigates Floer homotopy types within ample smooth divisor complements through the spectral lifting of the Ganatra-Pomerleano low-energy log PSS morphism. This approach enables the computation of the associated graded structure of these Floer homotopy types. A central aspect of this work involves the identification of an obstruction to the splitting process within the associated graded. This obstruction is characterized as a stable homotopy class, which is defined using higher-dimensional genus 0 relative Gromov-Witten moduli spaces.
Research Context
The study operates within the theoretical frameworks of Floer theory and Gromov-Witten theory, leveraging concepts such as homotopy types and moduli spaces. The Ganatra-Pomerleano low-energy log PSS morphism serves as a foundational element, which is then spectrally lifted to extend its applicability to the computation of Floer homotopy types.
Approach
The methodological core involves the spectral lifting of the Ganatra-Pomerleano low-energy log PSS morphism. This lifting procedure is applied to calculate the associated graded of Floer homotopy types. The research further defines an obstruction to splitting into the associated graded, situating this obstruction within the context of stable homotopy classes. These stable homotopy classes are formulated using higher-dimensional genus 0 relative Gromov-Witten moduli spaces.
Findings
- The Ganatra-Pomerleano low-energy log PSS morphism can be spectrally lifted to compute the associated graded of Floer homotopy types of ample smooth divisor complements.
- An obstruction exists to splitting into the associated graded, and this obstruction is encoded in a stable homotopy class.
- This stable homotopy class is defined via higher-dimensional genus 0 relative Gromov-Witten moduli spaces.
- Numerous examples of splittings were computed.
- These computed examples include the affine part of all smooth projective hypersurfaces of degree at least 2.