From Seiberg-Witten to Gromov: MCE and Singular Symplectic Forms

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 \cite{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 [T] 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.

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