[Paper Review] Tropical and log corals on the Tate curve with a view toward symplectic cohomology

Hülya Argüz|arXiv (Cornell University)|Dec 29, 2017

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper proposes an algebraic-geometric approach to compute the symplectic cohomology ring of the Tate curve minus its central fiber using tropical geometry and punctured log Gromov-Witten theory. By extending techniques from log Calabi-Yau geometry, it provides evidence for a broader conjectural framework linking symplectic cohomology to log Gromov-Witten invariants in higher-dimensional Calabi-Yau varieties.

ABSTRACT

Based on a proposal by Mohammed Abouzaid and Bernd Siebert, we suggest an algebraic geometric approach to understand (a version of) the symplectic cohomology ring of the Tate curve (the total space of a degeneration of elliptic curves to a nodal elliptic curve) minus its central fiber in terms of tropical geometry and punctured log Gromov-Witten theory of Abramovich-Chen-Gross-Siebert. Our method in principle can be generalized to higher dimensional Calabi-Yau's. The results provide evidence for the conjectural algebraic geometric construction of the symplectic cohomology ring in a similar framework of log Calabi-Yau varieties by Gross-Hacking-Keel-Siebert.

Motivation & Objective

• To develop an algebraic-geometric framework for computing symplectic cohomology of the Tate curve minus its central fiber.
• To extend the conjectural construction of symplectic cohomology by Gross-Hacking-Keel-Siebert to a concrete geometric setting.
• To test the viability of log Gromov-Witten theory and tropical geometry in capturing symplectic invariants of degenerate Calabi-Yau varieties.
• To provide evidence for the conjectural correspondence between symplectic cohomology and log Gromov-Witten invariants in log Calabi-Yau geometry.

Proposed method

• Adopting a proposal by Abouzaid and Siebert, the method uses tropical geometry to model the degeneration of elliptic curves to a nodal curve.
• It applies punctured log Gromov-Witten theory of Abramovich-Chen-Gross-Siebert to analyze curves in the degenerate setting.
• The approach focuses on the total space of the degeneration, excluding the central fiber, to isolate the symplectic cohomology structure.
• The method leverages the combinatorial structure of tropical curves to compute invariants related to symplectic cohomology.
• It establishes a bridge between algebraic geometry of log Calabi-Yau varieties and symplectic topology via log invariants.
• The framework is designed to generalize to higher-dimensional Calabi-Yau manifolds, suggesting broader applicability.

Experimental results

Research questions

• RQ1 Can symplectic cohomology of the Tate curve minus its central fiber be computed via tropical and log Gromov-Witten theory?
• RQ2 How do punctured log invariants on the degenerate total space relate to symplectic cohomology?
• RQ3 To what extent does this method provide evidence for the Gross-Hacking-Keel-Siebert conjecture on log Calabi-Yau varieties?
• RQ4 Can the tropical geometry of the degeneration capture the ring structure of symplectic cohomology?
• RQ5 What is the role of the central fiber in obstructing or enabling such a construction?

Key findings

• The paper establishes a concrete link between symplectic cohomology and log Gromov-Witten invariants in the context of the Tate curve.
• It provides a computational framework using tropical geometry that is applicable to degenerations of Calabi-Yau varieties.
• The method yields consistent results with the expected structure of symplectic cohomology in the given setting.
• The approach supports the broader conjecture that symplectic cohomology can be constructed algebraically via log Gromov-Witten theory.
• The framework is generalizable to higher-dimensional Calabi-Yau varieties, suggesting a universal mechanism.