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Cohomological localization for Hamiltonian $S^1$-actions and symmetries of complete intersections

Nicholas Lindsay

TL;DR

This work uses cohomological localization for Hamiltonian $S^1$-actions to relate global topological invariants to fixed-point data, extending Jones–Rawnsley and Farber results. It proves that eight-dimensional complete intersections with such actions must be diffeomorphic to specific multidegrees, removing the prior fixed-point-set assumptions, and provides a broad dimension-independent analogue under a similar fixed-point condition. The paper also develops a sharp GKM-based classification that ties infinite automorphism groups to the existence of certain torus actions and analyzes unimodality properties of even Betti numbers in low dimensions under hypotheses on the fixed-point set. Collectively, these results illuminate when symplectic manifolds with Hamiltonian symmetries resemble their Kähler or projective counterparts and offer tools for distinguishing symplectic rationality or monotone structures. The methods integrate localization formulas for Betti numbers and signature, Duistermaat–Heckman theory, and GKM combinatorics to constrain topology from symmetry.

Abstract

To begin the paper we revisit a cohomological localization result of Jones-Rawnsley which was subsequently improved by Farber, further generalizing the result. We then proceed to improve a previous result of the author on complete intersections of dimension $8k$ with a Hamiltonian $S^1$-action in two directions. Firstly, in dimension $8$ we remove the assumption on the fixed point set. Secondly, in any dimension we prove the result under an analogous assumption on the fixed point set. We also give some applications towards the unimodality of Betti numbers of symplectic manifolds having a Hamiltonian $S^1$-action, and discuss the relation to symplectic rationality problems.

Cohomological localization for Hamiltonian $S^1$-actions and symmetries of complete intersections

TL;DR

This work uses cohomological localization for Hamiltonian -actions to relate global topological invariants to fixed-point data, extending Jones–Rawnsley and Farber results. It proves that eight-dimensional complete intersections with such actions must be diffeomorphic to specific multidegrees, removing the prior fixed-point-set assumptions, and provides a broad dimension-independent analogue under a similar fixed-point condition. The paper also develops a sharp GKM-based classification that ties infinite automorphism groups to the existence of certain torus actions and analyzes unimodality properties of even Betti numbers in low dimensions under hypotheses on the fixed-point set. Collectively, these results illuminate when symplectic manifolds with Hamiltonian symmetries resemble their Kähler or projective counterparts and offer tools for distinguishing symplectic rationality or monotone structures. The methods integrate localization formulas for Betti numbers and signature, Duistermaat–Heckman theory, and GKM combinatorics to constrain topology from symmetry.

Abstract

To begin the paper we revisit a cohomological localization result of Jones-Rawnsley which was subsequently improved by Farber, further generalizing the result. We then proceed to improve a previous result of the author on complete intersections of dimension with a Hamiltonian -action in two directions. Firstly, in dimension we remove the assumption on the fixed point set. Secondly, in any dimension we prove the result under an analogous assumption on the fixed point set. We also give some applications towards the unimodality of Betti numbers of symplectic manifolds having a Hamiltonian -action, and discuss the relation to symplectic rationality problems.
Paper Structure (15 sections, 31 theorems, 55 equations)

This paper contains 15 sections, 31 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hamiltonian $S^1$-actions
  4. GKM actions
  5. Applications of Duistermaat-Heckman
  6. Complete Intersections
  7. Proof of Theorem \ref{['compin']}
  8. Proving the JR equation for $4$-dimensional fixed components
  9. Results in arbitrary dimensions
  10. Symplectic $S^1$-actions
  11. GKM actions
  12. Unimodality of Betti numbers
  13. Two technical lemmas regarding symplectic reduction
  14. Behavior of the reduced symplectic form near extremal values
  15. Behavior of the reduced symplectic form near certain fixed submanifolds

Key Result

Theorem 1.2

Suppose that $(M,\omega)$ is an $8$-dimensional complete intersection with a symplectic $S^1$-action. Then $M$ is diffeomorphic to one of $X_{4}(1),X_{4}(2)$ or $X_{4}(2,2)$.

Theorems & Definitions (57)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 47 more