Mayer-Vietoris systems and their applications

Abstract

We introduce notions such as cdp presheaf, cds precosheaf, Mayer-Vietoris system, and investigate their properties. As applications, we study cohomologies with values in local systems on smooth manifolds and Dolbeault cohomologies with values in locally free sheaves on complex manifolds, where the compactness is not necessary for both cases.In particular, we write out explicit blow-up formulas for these cohomologies. Moreover, we compare the blow-up formula given by Rao, S., Yang, S. and Yang, X.-D. with ours, and then deduce that their formula is still an isomorphism in the noncompact case.

The Double Complex of a Blow-up

“…As we have seen in the present article, the top map and the vertical ones are E 1 -isomorphisms. On the other hand, the results of [12] and [14] (c.f. also [4]) imply that the bottom map is an E 1 -isomorphism.…”

“…The isomorphism in the Dolbeault case is made explicit in [12] and the compactness hypothesis is removed. The method used there is conceptualized and extended in [14]. The Bott-Chern case is partially proved in [22] and complemented with a conjecture for the Bott-Chern cohomology of projective bundles (c.f.…”

On the Morse–Novikov Cohomology of blowing up complex manifolds

“…Remark. In [8], Meng studies the cohomologies with values in local system on smooth manifolds. Our approach should be applied to the blow-up formula for local system with finite rank.…”

“…The explicit morphism φ constructed here is inspired by [12], and owns an inverse direction to Meng's one. Actually in [8], Meng points out that these two morphism inverse to each other.…”

Dolbeault cohomologies of blowing up complex manifolds II: Bundle-valued case

“…In some sense, the morphism from the sheaf-theoretic approach is the inverse of the morphism from the topological approach. After our preprint had been distributed, Meng [29] informed us that he has also considered the Dolbeault blow-up formula for the bundle-valued case on non-compact manifolds by Mayer-Vietoris sequence.…”