The local Floer cohomology of indicator functions

Abstract

Let $K$ be a compact domain with contact-type boundary in a symplectic manifold $M$. We associate an unweighted local Floer cohomology group to each isolated Reeb component of $\partial K$ and show that, in sufficiently small Novikov windows, these groups together with $H^{\ast }(K)$ describe the symplectic cohomology of $M$ with support on $K$. The main analytic input is an energy-barrier construction which remains uniform under finite-slope approximations of the indicator Hamiltonian of $K$. We also study functoriality. For inclusions of contact-type domains we describe the local blocks of the restriction map without requiring the Liouville primitives on the two boundaries to agree. For a monotone isotopy carrying an isolated family of Reeb components, the corresponding block is the local Floer transport map multiplied by the Novikov weight determined by the source-minus-target action difference. These results provide the local Floer-theoretic input for companion work on relative symplectic cohomology near singularities of SYZ fibrations.

arXiv commentSubstantially revised and reorganized. The 4-dimensional SYZ singularity applications from v1 have been split off to a companion paper and replaced by an application to the regular locus in any dimension. The present version focuses on the local Floer-theoretic foundations, with strengthened functoriality and a direct small-window formulation of the main results