Descent of dg cohesive modules for open covers on complex manifolds

Abstract

In this paper we study the descent problem of cohesive modules on complex manifolds. For a complex manifold $X$ we could consider the Dolbeault dg-algebra $\mathcal{A}(X)$ on it and Block in 2006 introduced a dg-category $\mathcal{P}_{\mathcal{A}(X)}$, called cohesive modules, associated with $\mathcal{A}(X)$. The same construction works for any open subset $U\subset X$ and we obtain a dg-presheaf on $X$ given by $U\mapsto \mathcal{P}_{\mathcal{A}(U)}$. In this paper we prove that this dg-presheaf satisfies the descent property for any locally finite open cover of a complex manifold $X$. This generalizes part of the results of Ben-Bassat and Block in 2012, which studied the case that $X$ is covered by two open subsets.