Compact Lagrangians implied by noncompact mirror equivalence

Ascertain whether a homological mirror symmetry equivalence Fuk(X) = Coh(Y) for a Weinstein manifold X (with Y singular or noncompact) implies the existence of any compact Lagrangian submanifolds in X or in a compactification \overline{X}; specifically, determine if compact Lagrangian objects in the Fukaya category arise from or are guaranteed by this noncompact mirror equivalence.

Background

The paper discusses homological mirror symmetry in the Weinstein (noncompact) setting, where one has equivalences Fuk(X) = Coh(Y) with Y necessarily singular or noncompact. In many cases of interest, smoothing Y is expected to correspond to compactifying X.

However, the author emphasizes that the existence of compact Lagrangians does not automatically follow from such a noncompact mirror equivalence, highlighting a gap between the noncompact equivalence and compact objects in the Fukaya category.