From Seiberg-Witten to Gromov: MCE and singular symplectic forms
type: Item
instance of: scholarly article
title: From Seiberg-Witten to Gromov: MCE and singular symplectic forms
author: Yi-Jen Lee
published in: Journal of Differential Geometry
publication date: 2024-06-26T00:00:00Z
review text: Thanks to the work of C. Taubes [\textit{C. H. Taubes} and \textit{R. Wentworth} (ed.), Seiberg Witten and Gromov invariants for symplectic \(4\)-manifolds. Somerville, MA: International Press (2000; Zbl 0967.57001)], it is known that the Seiberg-Witten invariants of closed symplectic 4-manifolds correspond to certain counts of pseudo-holomorphic curves. In one direction, this is established by perturbing the Seiberg-Witten equations with a large multiple of the symplectic form, causing the solutions to converge to singular solutions supported on a union of connected pseudo-holomorphic curves with weights.\N\NFirst, building on a more general version of this result, which allows for harmonic perturbation terms vanishing along 1-dimensional submanifolds, this paper investigates solutions to the Seiberg-Witten equation on a closed \(\mathrm{Spin}^c\) \(3\)-manifold with \(b_1>0\), perturbed by a harmonic Morse-Novikov 1-form \(\theta\). Theorem 1.3 shows that a sequence of solutions, as the perturbation parameter tends to infinity, converges to the flow trajectories of the vector field dual to \(\theta\). The proof leverages Taubes's result applied to \(Y\times S^1\) and dimension reduction, i.e., comparing solutions on \(Y\) and \(Y\times S^1\). Theorem 1.3 represents a first step towards establishing a 3-dimensional version of the equivalence between Seiberg-Witten and Gromov-type invariants in dimension 3.\N\NThen the paper studies perturbed Seiberg-Witten equations on 4-manifolds with cylindrical ends, where each end is a closed 3-manifold times a half-line, equipped with a translation-invariant complete metric. For ``admissible'' perturbations, Theorems 1.8 and 1.10 show that a sequence of solutions, as the perturbation parameter tends to infinity, converges to singular solutions supported on a union of so-called t-curves.\N\NOverall, the motivations for these results include various conjectured connections between Seiberg-Witten-Floer homologies and Gromov-type …
reviewed by: Mohammad Farajzadeh Tehrani
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 57K41
zbMATH DE Number: 7871773
zbMATH Keywords: Seiberg-Witten invariants
zbMATH Keywords: Gromov-Witten invariants
zbMATH Keywords: Taubes's convergence theorem
MaRDI profile type: Publication
cites work: Singular Lefschetz pencils
cites work: Q3994678
cites work: Norms on the cohomology of hyperbolic 3-manifolds
cites work: Q3873183
cites work: Lefschetz pencils and the canonical class for symplectic four-manifolds
cites work: Compactness results in symplectic field theory
cites work: Embedded contact homology and its applications
cites work: Circle-valued Morse theory and Reidemeister torsion
cites work: Gluing pseudoholomorphic curves along branched covered cylinders. I
cites work: Proof of the Arnold chord conjecture in three dimensions. I
cites work: Proof of the Arnold chord conjecture in three dimensions. II
cites work: HF=HM. I: Heegaard Floer homology and Seiberg-Witten Floer homology
cites work: HF=HM. II: Reeb orbits and holomorphic curves for the ech/Heegaard Floer correspondence
cites work: HF=HM. III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence
cites work: HF=HM. IV: The Seiberg-Witten Floer homology and ech correspondence
cites work: HF=HM. V: Seiberg-Witten Floer homology and handle additions
cites work: Q5433081
cites work: Scalar curvature and the Thurston norm
cites work: Q5477303
cites work: Floer theoretic invariants for 3- and 4-manifolds
cites work: Periodic Floer homology and Seiberg-Witten-Floer cohomology
cites work: \(\underline{SW}\)=Milnor torsion
cites work: Lagrangian matching invariants for fibred four-manifolds. II.
cites work: Q3996430
cites work: Seiberg Witten and Gromov invariants for symplectic \(4\)-manifolds
cites work: Seiberg-Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2-forms
cites work: The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on \(S^1\times B^3\)
cites work: The Seiberg-Witten equations and the Weinstein conjecture
cites work: Q3709485
cites work: The Gromov invariant and the Donaldson-Smith standard surface count
cites work: Vortices and a TQFT for Lefschetz fibrations on 4-manifolds
DOI: 10.4310/JDG/1717772424
description: scientific article; zbMATH DE number 7871773