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

Kenneth Blakey

TL;DR

This work develops a spectral refinement of Floer theory for ample divisor complements by lifting the low-energy log PSS map to the level of Floer homotopy types and identifying an obstruction, $\mathcal{GW}$, built from genus-0 relative Gromov–Witten moduli. The vanishing of $\mathcal{GW}$ implies a wedge-splitting of the Floer homotopy type into Thom-spectra pieces; this provides a concrete criterion to compute the associated graded of the Floer spectrum. The authors execute numerous computations, including the affine parts of smooth projective hypersurfaces (and in particular the $T^*S^n$ and $T^*\mathbb{R}P^n$ cases), yielding explicit splittings and connecting to known algebro-geometric splittings of loop spaces. The framework interweaves symplectic cohomology, log cohomology, and spectral sequences with framed flow categories, offering computational pathways and posing conjectures about broader geometric realizations and further splittings beyond the examined examples.

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.

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

TL;DR

This work develops a spectral refinement of Floer theory for ample divisor complements by lifting the low-energy log PSS map to the level of Floer homotopy types and identifying an obstruction, , built from genus-0 relative Gromov–Witten moduli. The vanishing of implies a wedge-splitting of the Floer homotopy type into Thom-spectra pieces; this provides a concrete criterion to compute the associated graded of the Floer spectrum. The authors execute numerous computations, including the affine parts of smooth projective hypersurfaces (and in particular the and cases), yielding explicit splittings and connecting to known algebro-geometric splittings of loop spaces. The framework interweaves symplectic cohomology, log cohomology, and spectral sequences with framed flow categories, offering computational pathways and posing conjectures about broader geometric realizations and further splittings beyond the examined examples.

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.
Paper Structure (44 sections, 35 theorems, 399 equations, 13 figures)

This paper contains 44 sections, 35 theorems, 399 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Sneak peak of computations
  4. Questions, speculations, etc
  5. Recollections of the cohomological story, after Ganatra-Pomerleano
  6. Topological limit model for symplectic cohomology
  7. Log-smooth pairs
  8. Divisorially-adapted Floer data
  9. Symplectic cohomology
  10. Log cohomology
  11. The spectral sequence
  12. The low-energy log PSS morphism
  13. Rapid review of Floer homotopy on Liouville manifolds
  14. Morse homotopy
  15. Spectral symplectic cohomology
  16. ...and 29 more sections

Key Result

Theorem 1.3

Assume Assumption assu:main. There exists an element referred to as a spectral Gromov-Witten obstruction, with the following property: suppose $\mathcal{G}\mathcal{W}=0$, then $\mathfrak{F}^\Lambda$ splits as a wedge sum of Spanier-Whitehead duals of Thom spectra:

Figures (13)

  • Figure 1: This is a $k$-marked thimble, $k\geq1$. The arrow at $z_1$ indicates our auxiliary choice of real tangent ray made in order to define the enhanced evaluation map; in future figures, we will omit this choice from the picture.
  • Figure 2: This is a broken (low-energy) $k$-marked thimble. Note, each $y_j$ has winding number equal to $x$'s winding number.
  • Figure 3: This is a (low-energy) hybrid 0-marked thimble; here, $x$ has winding number 0.
  • Figure 4: This is a (low-energy) hybrid $k$-marked thimble, $k\geq1$; here, $x$ has winding number $k$. Technically, the curve going into $a$ should lie in $S_DM$.
  • Figure 5: This is a schematic picture of the proof of Proposition \ref{['prop:he']}. In particular, the degenerations of the middle picture, with respect to the parameter $r$ measuring the length of the shaded region, to the top and bottom pictures depicts the commutativity of the diagram \ref{['eqn:commutativityfig']}; moreover, the "equality" in the bottom picture depicts the equality \ref{['eqn:equalityfig']}, i.e., the fact that GP's low-energy log PSS morphism is an isomorphism of rings. Note, on the left side of the "equality", each of the $\overline{w}(k)$ intersection points has multiplicity 1; meanwhile, on the right side of the "equality", the single intersection point has multiplicity $\overline{w}(k)$.
  • ...and 8 more figures

Theorems & Definitions (115)

  • Remark 1.1: Positivity of intersection
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Conjecture 1.10
  • ...and 105 more