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Vanishing of Rabinowitz Floer homology on negative line bundles

Peter Albers, Jungsoo Kang

TL;DR

The paper constructs Rabinowitz Floer homology for negative line bundles over symplectic bases under a semi-positivity assumption and proves vanishing results in several natural regimes, highlighting that SH need not vanish in these settings. It develops a carefully restricted transversality framework and a Morse-Bott perturbation to define RFH, then employs a filtration to reduce the vanishing problem to fiberwise Floer trajectories, which vanish due to known RFH calculations on $S^1$-fibers. The results elucidate a nuanced distinction between RFH and SH beyond the symplectically aspherical case and suggest a conceptual explanation via the CFO-Oancea long exact sequence, with concrete implications for examples like negative line bundles over projective spaces. The work broadens RFH construction to settings with holomorphic spheres and provides a concrete mechanism for when RFH must vanish, enriching the landscape of symplectic invariants for negative line bundles.

Abstract

Following [Fra08, AF14] we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. In [Rit14] Ritter showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem $\mathrm{SH}=0\Leftrightarrow\mathrm{RFH}=0$, [Rit13], does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak-Frauenfelder-Oancea long exact sequence [CFO10].

Vanishing of Rabinowitz Floer homology on negative line bundles

TL;DR

The paper constructs Rabinowitz Floer homology for negative line bundles over symplectic bases under a semi-positivity assumption and proves vanishing results in several natural regimes, highlighting that SH need not vanish in these settings. It develops a carefully restricted transversality framework and a Morse-Bott perturbation to define RFH, then employs a filtration to reduce the vanishing problem to fiberwise Floer trajectories, which vanish due to known RFH calculations on -fibers. The results elucidate a nuanced distinction between RFH and SH beyond the symplectically aspherical case and suggest a conceptual explanation via the CFO-Oancea long exact sequence, with concrete implications for examples like negative line bundles over projective spaces. The work broadens RFH construction to settings with holomorphic spheres and provides a concrete mechanism for when RFH must vanish, enriching the landscape of symplectic invariants for negative line bundles.

Abstract

Following [Fra08, AF14] we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. In [Rit14] Ritter showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem , [Rit13], does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak-Frauenfelder-Oancea long exact sequence [CFO10].

Paper Structure

This paper contains 8 sections, 14 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Rabinowitz Floer homology and Hamiltonian Floer homology
  3. Preliminaries
  4. Transversality
  5. Compactness
  6. Rabinowitz Floer homology
  7. A filtration and the proof of vanishing
  8. A conjectural explanation

Key Result

Theorem 1.1

The Rabinowitz Floer homology $\mathrm{RFH}(\Sigma_\tau,E)$ is well-defined for all $\tau>0$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.6
  • Lemma 2.7
  • ...and 34 more