Global mirror symmetry via descent for symplectic cohomology

Let $M$ be a closed graded manifold with a weakly involutive cover $M=\bigcup_{i=1}^N C_i$, and let $Y=\bigcup_{i=1}^N Y_i$ be a smooth rigid analytic space over $\Lambda$ with an admissible affinoid cover and a global non-vanishing section of its canonical bundle. For each non-empty $J\subset\{1,\ldots,N\}$, write $Y_J=\bigcap_{m\in J}Y_m$, let $\mathcal{O}_J$ be the algebra of functions on $Y_J$, and let $\operatorname{Der}(\mathcal{O}_J)$ be the Lie algebra of $\Lambda$-linear derivations of $\mathcal{O}_J$. The sheaf $\bigwedge TY$ consists of polyvector fields on $Y$. Assume that, for every such $J$, there is a $BV_\infty$ quasi-isomorphism of homotopy BV-algebras

SC^*_{M,\big}\left(\bigcap_{m\in J}C_m;\Lambda\right)\to \operatorname{Sym}^*_{\mathcal{O}_J}\left(\operatorname{Der}(\mathcal{O}_J)[1]\right)

compatible with restriction maps. Global mirror symmetry conjecture. Under these assumptions, there is a $BV_\infty$ quasi-isomorphism of homotopy BV-algebras

SC^*_{M,\big}(M;\Lambda)\simeq C^*(M;\Lambda)\to TW(Y,\bigwedge TY),

where $TW$ denotes the Thom–Whitney construction for the cover $Y=\bigcup_{i=1}^N Y_i$. This conjectural statement proposes that compatible local mirror-symmetry quasi-isomorphisms descend to a global one; the paper presents the section as part of a broader conjectural program rather than establishing the assertion.