Let $T = \mathrm{Spec}\ \mathbb{C}[x_1^{\pm 1}, x_2^{\pm 1}, \ldots, x_n^{\pm 1}]$ be an algebraic torus and $X$ a closed subvariety. Let $\eta$ be a differential form on $T$ of …

Let $p$ be a prime number. Let $X\to\operatorname{Spec}\mathbb Z_p$ be smooth and proper. Is it true that the map $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bu …

Hi, Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular …

Suppose $M \xrightarrow{p} X$ is a surjective submersion of smooth manifolds. Define the fibered product in this case: $$M \times_X M = \{ (m_1, m_2) \in M \times M | p(m_1) = p(m …

(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) …

In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M. And in the page: http://mathoverflow.net/questions/111059/relative-de-rham-cohomologies/ …

Let $\omega = \mathrm d \eta$ be an exact rational $n$-form on $\Bbb P^n$. It may happen that the polar locus of $\eta$ is not included in the polar locus of $\omega$. But is it …

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 topo …

I'd really like to understand the proof that Paul Monsky wrote about the finiteness of the de Rham cohomology of algebraic varieties. I'd like it very much because it seems to expl …

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 $\overline{\partial}$ …

Given a smooth hypersurface $X \subset \mathbb{P}^{2p+1}_{\mathbb{C}}$ of degree $d \ge 5$ (with $p\ge 2$), when can we say that $H^{p+1}(\mathcal{O}_X)=H^{p+1}(\Omega^1_X)=...=H …

In an answer to a previous question I asked, where the Kahler differentials of a variety over a finite field were discussed, it was stated that: You can certainly define de Rha …

I have a question on something that is probably very basic. Why is the de Rham cohomology of a single point equal to R?

Suppose that $Y/k$ is a an algebraic variety over a field $k$ of characteristic zero and that $Y\subseteq X$ is a closed embedding into a smooth variety over $k$. Then the completi …