Recently, while delving into the literature, I encountered two puzzling questions regarding complex manifolds.
part 1:
In defining the topological characteristics of a complex manifold of complex dimension (n) in terms of the cohomology group, which of the following is the correct formula? Is it $X(M)=\sum_{i = 0}^n (-1)^i \dim H^i(M,\mathbb{C})$ or $(X(M)=\sum_{i = 0}^{2n} (-1)^i \dim H^i(M,\mathbb{C})?$
Concerning the Grothendieck vanishing theorem, it states that $H^i(\mathbb{CP}^n,\mathbb{C}) = 0 \quad for \quad i > n$. However, we also know that $H^{2n}(\mathbb{CP}^n,\mathbb{C})\neq 0$. What seems to be the discrepancy here? Could it be related to the Zariski topology? Moreover, are these two cohomologies (the one in the vanishing theorem context and the one with the non-zero $H^{2n} $ isomorphic in some sense?
I would greatly appreciate any insights or clarifications on these points.
Part 2:
"We often encounter the definition of the étale Euler characteristic as a sum involving étale cohomology groups:
$$\chi_{\text{ét}}(X) = \sum_{i=0}^{\infty} (-1)^i \text{dim}_{\mathbb{Q}_\ell} H^i_{\text{ét}}(X, \mathcal{F})$$
where $H^i_{\text{ét}}(X, \mathcal{F})$ represents the i-th étale cohomology group of a scheme X with coefficients in a suitable sheaf $\mathcal{F}$.
However, I'm struggling to fully grasp the concept of étale cohomology, particularly when the coefficients are in a constant sheaf. What exactly does it mean to have a constant sheaf in the context of étale cohomology, and how does it relate to this definition of the Euler characteristic?"
