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### Relative De Rham cohomologies

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) ...
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### Monsky's proof of the finiteness of de Rham cohomology

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 explain in concrete ...
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### 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 ...
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### Is there a Hodge isomorphism theorem for part-tangential, part-normal, harmonic differential forms?

Let $M$ be an oriented compact Riemannian $n$-manifold with boundary $\partial M$. A differential $p$-form $\omega$ on $M$ is normal if $i^* \omega = 0$ holds, tangential if $i^* \star \omega = 0$ ...
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### Steenrod operations in algebraic geometry

What are some applications of Steenrod operations (or similar constructions) in algebraic geometry? I am dimly aware of the the use of these Voevodsky's work on motivic cohomology, and would be ...
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### Proof of Lefschetz-Hopf Fixpoint Theorem with de Rham cohomology?

Looking for a proof of the Lefschetz-Hopf Fixpoint Theorem with the de Rham Cohomology. (I´m more interestet in the Formula then just the simple statement that if the Lefschetz number is not zero ...
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 true that $\omega = \... 2answers 587 views ### If a$d \log$form is exact, is it zero? Let$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 the form $$\sum_I ... 1answer 422 views ### algebraic de rham cohomology of a curve Let X be a smooth projective curve over a field k of characteristic zero. The algebraic de Rham cohomology of X is, by definition, the hypercohomology of the complex of Kähler differentials for ... 1answer 472 views ### algebraic de Rham cohomology of singular varieties 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 cohomology H^*_{... 0answers 282 views ### Hard Lefschetz in De Rham cohomology I'm looking for a reference for Hard Lefschetz theorem in algebraic De Rham cohomology. By this I mean the statement that If i: Y \hookrightarrow X is a smooth hyperplane section of a smooth ... 2answers 365 views ### De Rham isomorphism for noncompact manifolds? Maybe someone has a quick answer. Thanks. For noncompact manifolds, is the De Rham cohomology isomorphic to the singular cohomology? Is the De Rham cohomology defined with the cochain of compactly ... 1answer 325 views ### Integration currents VS Poincaré Dual Let M be a complex manifold of dimension n and S \subset M a closed complex submanifold of complex codimension r. Let [S] \in H_{2r}(S) be the fundamental class of S. We have the ... 1answer 190 views ### Some clarifications regarding Deligne's paper on \ell-adic representations arising from modular forms I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ... 0answers 155 views ### Hodge filtration over \mathbb Z_p 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^\bullet_{X/\mathbb Z_p})... 0answers 384 views ### Finite Field Varieties and the de Rham Complex of Kähler Differentials 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 Rham cohomology using ... 2answers 602 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 ... 1answer 414 views ### Homology of a region of the plane This is related to this MO question, I don't know if it's really "research-level". As in that question, let U be a domain of the complex plane \mathbb{C}, i.e. an open connected subset. Let$$ \... 1answer 409 views ### algebraic de Rham cohomology functoriality 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 completion of the de Rham ... 0answers 288 views ### Gauss Manin connection in algebraic geometry and DG setting E.Geltzler defined Gauss Manin connection on periodic cyclic homology of smooth proper DG algebra.I just began to learn his definition.But it seems that his construction is very different to the Gauss ... 1answer 340 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$by the following ... 1answer 210 views ### Most general “finiteness of de Rham cohomology” statement for holonomic$D$-modules in the algebraic case? Let$X$be a nonsingular algebraic variety over a field$k$of characteristic zero. (We may assume$k$algebraically closed if need be, but I want to avoid specifically demanding$k = \mathbb{C}$.) ... 0answers 70 views ### Compact Vertical Cohomology and Euler Class of CP1 First of all, please excuse my English. I'm not native Englsih speaker, so you will see so many grammer mistakes, I just only hope that my mistkae wouldn't effect what I want to mean. Hi, recently I'... 0answers 79 views ### Reference: Relative cohomology of a morphism Let$f\colon Y \to X$be a morphism of schemes, the inverse image in$K$-theory always fit into a long exact sequence $$\cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots$$ where the ... 0answers 179 views ### Mayer-Vietoris on Fibered Products Suppose$M \xrightarrow{p} X$is a surjective submersion of smooth manifolds. Define the fibered product in this case: $$M \times_X M = \{ (m_1, m_2) \in M \times M | p(m_1) = p(m_2) \}$$ and let$U =...
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^{p+1}(\Omega^{p-2}...