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Maurer--Cartan elements in symplectic cohomology from compactifications

Matthew Strom Borman, Mohamed El Alami, Nick Sheridan

Abstract

We prove that under certain conditions, a normal crossings compactification of a Liouville domain determines a Maurer--Cartan element for the $L_\infty$ structure on its symplectic cohomology; and deforming by this element gives the quantum cohomology of the compactification.

Maurer--Cartan elements in symplectic cohomology from compactifications

Abstract

We prove that under certain conditions, a normal crossings compactification of a Liouville domain determines a Maurer--Cartan element for the structure on its symplectic cohomology; and deforming by this element gives the quantum cohomology of the compactification.
Paper Structure (28 sections, 25 theorems, 206 equations, 2 figures)

This paper contains 28 sections, 25 theorems, 206 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometric setup
  3. Maurer--Cartan element
  4. Deforming by the Maurer--Cartan element
  5. Conjectures
  6. Implications for mirror symmetry in the log Calabi--Yau case
  7. Outline
  8. Geometric Background
  9. Quantum Cohomology
  10. Simple crossings divisors
  11. The Liouville structure
  12. When $D$ is smooth
  13. When $D$ is orthogonal simple crossings
  14. The radial coordinate
  15. Action-Index computations
  16. ...and 13 more sections

Key Result

theorem 1

Under Hypothesis hyp, we have an isomorphism of graded $R$-modules

Figures (2)

  • Figure 1: Linear Hamiltonian, with the points $1$, $\sigma_+ = 1-\eta$, $\sigma = 1-2\eta$, $\sigma_- = 1-3\eta$, $1-4\eta$, and $1-5\eta$ marked on the $\rho$-axis.
  • Figure 2: Pruning weight $0$ internal edges of a tree.

Theorems & Definitions (55)

  • theorem 1
  • definition 1
  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • lemma 1
  • corollary 1
  • lemma 2
  • ...and 45 more