From Seiberg-Witten to Gromov: MCE and singular symplectic forms
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 Floer homologies.
scientific article; zbMATH DE number 7871773
scientific article