Disjoinable Lagrangian tori and semisimple symplectic cohomology

TL;DR

This paper establishes constraints on Lagrangian embeddings in certain symplectic manifolds with semisimple symplectic cohomology, linking geometric surgeries to algebraic properties of Fukaya categories.

Contribution

It introduces new constraints on Lagrangian embeddings in symplectic fillings with semisimple cohomology and relates these to generalized boundary connected sums and birational surgeries.

Findings

Manifolds with semisimple symplectic cohomology can be constructed via boundary connected sums.

Such manifolds exhibit proper wrapped Fukaya categories.

The results imply the existence of many non-toric monotone symplectic manifolds with proper Fukaya categories.

Abstract

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-downs and flips. As a consequence, there are many non-toric (non-compact) monotone symplectic manifolds whose wrapped Fukaya categories are proper.