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

Disjoinable Lagrangian tori and semisimple symplectic cohomology