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Degenerate symplectic fixed points and Gromov-Witten invariants

Wenmin Gong

TL;DR

The work builds a bridge between degenerate symplectic fixed points and Gromov–Witten invariants by developing ζ-deformed spectral invariants and big quantum homology. It extends Givental’s fixed-point theorem to closed rational symplectic manifolds using nonzero mixed GW invariants with fixed markings, via deformed quantum cuplength and principal quantum factorizations. The paper introduces deformed Floer theory, PSS isomorphisms, and spectral invariants, then derives concrete fixed-point lower bounds for Hamiltonian diffeomorphisms, including monotone cases and toric examples, with a broad set of corollaries for uniruled, PFQF-bearing, and blown-up manifolds. This framework yields practical lower bounds on the number of fixed points and connects symplectic dynamics to GW theory across a range of geometries, including Fano toric manifolds and projective bundles.

Abstract

We establish a connection between Gromov-Witten invariants and the number of fixed points of Hamiltonian diffeomorphisms on a closed rational symplectic manifold via deformed Hamiltonian spectral invariants. We generalize Givental's symplectic fixed point theorem for Fano toric manifolds to closed rational symplectic manifolds which admit nonzero Gromov-Witten invariants with fixed marked points and one point insertion. We prove a new cuplength estimate of symplectic fixed points involved in deformed spectral invariants. We extend Schwarz's quantum cuplength to the notion of deformed quantum cuplength for symplectic periods and employ it to estimate the number of fixed points of Hamiltonian diffeomorphisms on monotone symplectic manifolds with nonzero mixed Gromov-Witten invariants.

Degenerate symplectic fixed points and Gromov-Witten invariants

TL;DR

The work builds a bridge between degenerate symplectic fixed points and Gromov–Witten invariants by developing ζ-deformed spectral invariants and big quantum homology. It extends Givental’s fixed-point theorem to closed rational symplectic manifolds using nonzero mixed GW invariants with fixed markings, via deformed quantum cuplength and principal quantum factorizations. The paper introduces deformed Floer theory, PSS isomorphisms, and spectral invariants, then derives concrete fixed-point lower bounds for Hamiltonian diffeomorphisms, including monotone cases and toric examples, with a broad set of corollaries for uniruled, PFQF-bearing, and blown-up manifolds. This framework yields practical lower bounds on the number of fixed points and connects symplectic dynamics to GW theory across a range of geometries, including Fano toric manifolds and projective bundles.

Abstract

We establish a connection between Gromov-Witten invariants and the number of fixed points of Hamiltonian diffeomorphisms on a closed rational symplectic manifold via deformed Hamiltonian spectral invariants. We generalize Givental's symplectic fixed point theorem for Fano toric manifolds to closed rational symplectic manifolds which admit nonzero Gromov-Witten invariants with fixed marked points and one point insertion. We prove a new cuplength estimate of symplectic fixed points involved in deformed spectral invariants. We extend Schwarz's quantum cuplength to the notion of deformed quantum cuplength for symplectic periods and employ it to estimate the number of fixed points of Hamiltonian diffeomorphisms on monotone symplectic manifolds with nonzero mixed Gromov-Witten invariants.

Paper Structure

This paper contains 26 sections, 22 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Applications and examples
  3. The quadric in ${\mathbb{P}}^{n+1}$
  4. Fano toric manifolds
  5. $\mathbb{P}^n$ blown up at one point
  6. Projective bundles over ${\mathbb{P}}^n$
  7. Organization of the paper
  8. Preliminaries
  9. Notations and conventions
  10. Big quantum homology ring
  11. Deformed quantum cuplength
  12. Morse theoretical big quantum homology
  13. Deformed Hamiltonian Floer homology
  14. Deformed Floer complexes
  15. Gradings
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(M^{2n},\omega)$ be a closed rational symplectic manifold with minimal symplectic period $p>0$ which admits a nonzero mixed Gromov-Witten invariant of the form: where $A\in H_2(M)$, $a_1,\ldots, a_l\in H_{*<2n}(M;{\mathbb{Q}})$ with $2\leq l\in{\mathbb{N}}$, and the classes $b_1,\ldots,b_k$ are rational homology classes of even degree. If where the infimum is taken over any $c\in H_*(M;{\ma

Theorems & Definitions (46)

  • Theorem 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5: Sc2
  • Definition 1.6
  • ...and 36 more