Descent of dg cohesive modules for open covers on compact complex manifolds

Abstract

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

A recurrent formula of $A_{\infty}$-quasi inverses of dg-natural transformations between dg-lifts of derived functors

“…In [Wei19] the formula in this note is used to construct A ∞ -inverse of dg-natural transformations between twisted complexes. Moreover in an upcoming work [Wei], the formula in this note, together with the method in [Wei16], [Wei18], [BHW17], and [AØ18], can be used to obtain an injective dg-resolution of the equivariant derived category [BL94].…”

Twisted complexes and simplicial homotopies

“…Proposition 1.6 shows the importance of twisted complexes in descent theory, See [AØ18, Introduction] for some discussions and [Wei18] for an application.…”

The descent of twisted perfect complexes on a space with soft structure sheaf

“…Remark 1. Part of this paper, in particular Section 3.2, has been integrated in to [Wei18], although the notations and viewpoint are slightly different.…”