From Seiberg-Witten to Gromov: MCE and singular symplectic forms

Abstract

Motivated by various possible generalizations of Taubes’s $SW = Gr$ theorem [T] to Floer-theoretic setting, we prove certain variants of Taubes’s convergence theorem in [T] (the first part of his proof of $SW = Gr$). In place of the closed symplectic $4$-manifold considered in [T], this article considers non-compact manifolds with cylindrical ends, equipped with a self-dual harmonic $2$-form with non-degenerate zeroes. This extends and simplifies some central technical ingredients of the author’s prior work in [LT] and [KLT5]. Other expected applications include: extending the $HM = PFH$ theorem in [LT] and the $HM = HF$ theorem in [KLT1]–[KLT5] to TQFTs on both sides [L1]; definitions of large-perturbation Seiberg–Witten analogs of Heegaard Floer theory’s link Floer homologies and link cobordism invariants.