Abouzaid’s toric line-bundle theorem is a partial result

For the generic source Laurent potential and selected section data of Theorem 1.2, Abouzaid identifies a full sub-pre-category of the relative Fukaya pre-category with the dg category of line bundles on a smooth projective toric variety. The source then uses a separate generation result for coherent sheaves by line bundles to obtain an embedding of $D^b\mathrm{Coh}(X)$ into a derived Fukaya category. It does not prove essential surjectivity onto the whole Fukaya category and therefore is not full HMS. This boundary matters especially outside the Fano case: the source notes that other bounded components of the complement of the tropical amoeba can contribute sections with no corresponding coherent-derived object. The theorem must retain its line-bundle and selected-section scope.