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Index bounded relative symplectic cohomology

Yuhan Sun

TL;DR

This work develops a computational framework for relative symplectic cohomology $SH_M(D)$ of a Liouville domain $D$ in a Calabi–Yau manifold $M$ by introducing index-bounded boundary conditions and a filtration on the completed Floer telescope. It constructs a spectral sequence that starts at the classical symplectic cohomology $SH(D;\Lambda_E)$ and converges to $SH_M(D;\Lambda_E)$; in the integral lift case, the spectral sequence collapses when $[\tilde{\omega}]$ vanishes on $H_2(M,D)$. The approach leverages lower semi-continuous Hamiltonians, the $S$-shape Hamiltonians in cylindrical regions, and an Abouzaid–Seidel-type maximum principle to isolate contributions from inside $D$, enabling explicit computations and applications such as non-displaceability results for Weinstein neighborhoods of simply-connected Lagrangians. The framework also extends to perturbations of Morse–Bott Reeb orbits, providing a robust method to handle degenerate boundary dynamics and to study local-to-global displacement phenomena in Calabi–Yau settings.

Abstract

We study the relative symplectic cohomology with the help of an index bounded contact form. For a Liouville domain with an index bounded boundary, we construct a spectral sequence which starts from its classical symplectic cohomology and converges to the relative symplectic cohomology of it inside a Calabi-Yau manifold.

Index bounded relative symplectic cohomology

TL;DR

This work develops a computational framework for relative symplectic cohomology of a Liouville domain in a Calabi–Yau manifold by introducing index-bounded boundary conditions and a filtration on the completed Floer telescope. It constructs a spectral sequence that starts at the classical symplectic cohomology and converges to ; in the integral lift case, the spectral sequence collapses when vanishes on . The approach leverages lower semi-continuous Hamiltonians, the -shape Hamiltonians in cylindrical regions, and an Abouzaid–Seidel-type maximum principle to isolate contributions from inside , enabling explicit computations and applications such as non-displaceability results for Weinstein neighborhoods of simply-connected Lagrangians. The framework also extends to perturbations of Morse–Bott Reeb orbits, providing a robust method to handle degenerate boundary dynamics and to study local-to-global displacement phenomena in Calabi–Yau settings.

Abstract

We study the relative symplectic cohomology with the help of an index bounded contact form. For a Liouville domain with an index bounded boundary, we construct a spectral sequence which starts from its classical symplectic cohomology and converges to the relative symplectic cohomology of it inside a Calabi-Yau manifold.

Paper Structure

This paper contains 16 sections, 32 theorems, 124 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Hamiltonian Floer theory on closed manifolds
  4. Liouville domain and contact cylinder
  5. Relative symplectic cohomology of a Liouville domain
  6. Hamiltonian Floer theory on manifolds with convex boundary
  7. Lower semi-continuous Hamiltonian functions
  8. A filtration on the completed telescope
  9. Auxiliary symplectic forms
  10. A filtration which is not exhaustive
  11. The induced spectral sequence
  12. Ignoring upper orbits and convergence
  13. The first page of the spectral sequence
  14. Examples and extensions
  15. Simply-connected Lagrangians in Calabi-Yau manifolds
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M, \omega)$ be a closed symplectic Calabi-Yau manifold and $D$ be a Liouville domain in $M$ with an index bounded boundary. Suppose that $[\tilde{\omega}]$ is integral. Given any positive number $E$, there is a truncated invariant $SH_{M}(D; \Lambda_{E})$ such that

Figures (4)

  • Figure 1: Hamiltonian functions in the cylindrical coordinate.
  • Figure 2: A sandwich of Hamiltonian functions
  • Figure 3: Cut-off functions for $\tilde{\omega}$ and $\tilde{\theta}$.
  • Figure 4: Hamiltonian functions with fixed lower parts and small variations outside

Theorems & Definitions (65)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • ...and 55 more