0
votes
0answers
72 views

Comparison of support of a divisor and its class in de Rham cohomology

Let $X$ be a smooth surface over $\mathbb{C}$ and $D$ be an effective divisor. Suppose that the linear system corresponding to $D$ is zero dimensional. Denote by $[D] \in H^2(X,\mathbb{Z})$ the ...
1
vote
1answer
234 views

deRham cohomology of a manifold with covering space $S^{n}$

(A qual problem) Let $\pi:S^{n}\rightarrow M$ be a covering map, $M$ being an orientable manifold. Show that $H_{deR}^{k}(M)=0$ for $1\leq k < n $. We can show $H_{deR}^{1}(M)=0$ by the following ...
4
votes
1answer
454 views

de rham model for relative cohomology

In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M. And in the page: Relative De Rham cohomologies , I got to know that there is another model for relative ...
9
votes
1answer
645 views

Relative De Rham cohomologies

Hello, as far as I know, there are two main ways to have a relative version of De Rham Cohomology for a pair (M,N), where M and N are smooth manifolds and N is a closed (as a topological subspace) ...
2
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2answers
451 views

Global Definition of the Dolbeault Complex of a Vector Bundle

For an $2n$-dimensional complex manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator $\overline{\partial}$, built locally from ...