Sectorial covers over fanifolds
Hayato Morimura
TL;DR
The paper addresses HMS for fanifolds by constructing a Weinstein sectorial cover of the stopped sector $\widetilde{\mathbf{W}}(\Phi)$ and showing HMS as a cosheaf isomorphism, thus gluing local (toric) data to a global equivalence. The main method combines inductive Weinstein handle attachments with a sectorial descent framework to produce three equivalent cosheaves on the fanifold and to equate the A- and B-model sides via $\bar{\pi}_* \operatorname{Fuk}_* \cong \operatorname{Coh}_! \circ \mathbf{T}$. The key result is a global HMS equivalence $\operatorname{Fuk}(\widetilde{\mathbf{W}}(\Phi)) \simeq \operatorname{Coh}(\mathbf{T}(\Phi))$, and its refinement for very affine hypersurfaces shows the GS2 conjecture holds by lifting the open skeleton cover to a Weinstein sectorial cover. This work thus provides a robust, geometry-driven path to HMS in the fanifold setting, unifying $A$- and $B$-model perspectives through cosheaf-theoretic glueing and sectorial descent with explicit toric and FLTZ ingredients.
Abstract
For the stopped Weinstein sector associated with any fanifold recently introduced by Gammage--Shende, we construct a Weinstein sectorial cover which allows us to describe homological mirror symmetry over the fanifold as an isomorphism of cosheaves of categories. In a special case, our Weinstein sectorial cover gives a lift of the open cover for the global skeleton of a very affine hypersurface computed in their previous work.