Twisted polytope sheaves and coherent–constructible correspondence for toric varieties

Abstract

Given a smooth projective toric variety $$X_\Sigma $$ of complex dimension n, Fang–Liu–Treumann–Zaslow (Invent Math 186(1):79–114, 2011) showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $$Coh(X_\Sigma )$$ into the dg derived category of constructible sheaves on a torus $$Sh(T^n, \Lambda _\Sigma )$$ . Recently, Kuwagaki (The nonequivariant coherent-constructible correspondence for toric stacks, 2016. arXiv:1610.03214 ) proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors.