Compact Lagrangians from noncompact homological mirror symmetry

Ascertain whether, given a homological mirror symmetry equivalence Fuk(X) ≅ Coh(Y) for a Weinstein symplectic manifold X (with Y singular or noncompact), it follows that X or a compactification \overline{X} admits any compact exact Lagrangian submanifolds.

Background

The authors discuss homological mirror symmetry in the noncompact/Weinstein setting, where one often has an equivalence Fuk(X) ≅ Coh(Y). While smoothing Y is expected to correspond to compactifying X, it is unclear whether such an equivalence guarantees the existence of compact Lagrangians in X or its compactification.

This issue is important because compact Lagrangians are the relevant objects when attempting to pass from noncompact to compact mirror symmetry statements.