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1
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2answers
425 views

Hyper(co)homology of exact (acyclic) complexes

Let $\mathcal{A}$ be an abelican category with enough injectives, let $K^\bullet \in Kom^+(\mathcal{A})$ be a complex, where $Kom^+(\mathcal{A})$ is the category of cochain complexes over ...
6
votes
1answer
268 views

Effective Serre Vanishing

Suppose that $X = \mathbb{P}^n_k$ and $G$ is a coherent sheaf on $X$. Question: Is there a way to determine some integer $n_0$ such that $H^1(X, G \otimes O_X(n)) = 0$ for all $n \geq n_0$? ...
10
votes
1answer
357 views

Wanted: Odd-dimensional integral cohomology class whose square is nonzero

Does anyone know of a nice simple example of a space $X$ with an odd-dimensional integral cohomology class $a\in H^{2k+1}(X;\mathbb{Z})$ whose square is nonzero? I once thought that the ...
2
votes
1answer
254 views

Cohomology groups of quotient by finite group

I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question. Let $G$ be a finite group acting on a manifold $M$ without fixed ...
8
votes
1answer
836 views

Differential Calculus and the De Rham Homotopy Operator

Suppose $M$ is a smooth manifold, $\Omega^k(M)$ the vector space (or $C^\infty(M)$-module in case this is better suited) of $k$-differentialforms on $M$ and $$I: \Omega^k(M\times \mathbb{R}) \to ...
1
vote
0answers
317 views

Vanishing of cohomology groups

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 ...
6
votes
1answer
283 views

Torsion in $H^2(X,\mathbb{Z})$ induced by torsion in $H_1(X,\mathbb{Z})$

Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $$ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow ...
0
votes
1answer
488 views

Infinite linear span vs closed linear span

Hi, Suppose we have a (real, separable) Banach space $V$ and a (linear) set $A\subseteq V$. I presume in general it might not be possible to write every element of the closed span of $A$ as an ...
5
votes
2answers
254 views

cohomology and $j_!$

I have a projective variety $X$ and an open immersion $j : U \to X$. Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between ...
2
votes
0answers
238 views

Defining the cup product in Ext using a Kunneth formula

I want to make a Kunneth product of sorts on Ext. In particular, letting $C_\*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represented by maps in ...
2
votes
2answers
468 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 ...
6
votes
1answer
542 views

Equivalence between statements of Hodge conjecture

Dear everyone, I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has ...
3
votes
0answers
366 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 ...
0
votes
2answers
375 views

Reference for equivariant Riemann-Roch formula?

Is there any reference for equivariant Riemann-Roch formula: book, paper, notes or something? I want to compute the weight of the action of C^* on the top wedge of cohomology group.
2
votes
1answer
356 views

equivariant cohomology

In this question we consider only cohomology with rational coefficients. All groups will be connected Lie groups. All group actions will be smooth. Let $M$ be a manifold. Let $G$ be a group acting on ...
2
votes
1answer
367 views

system of local coefficients on X, locally constant sheaves and orientation sheaves

Hi, I try to understand the orientation sheaves. When searching it in the google, i meet new areas such as local coefficient system and locally constant sheaves. I realize that any system of local ...
3
votes
0answers
190 views

Cohomologies of polyvector fields of a toric variety.

Hello. I am sorry if this is a totally trivial question. I am interested in computing cohomologies of polyvector fields of an arbitrary complete (algebraic) toric variety (assume smooth if otherwise ...
18
votes
2answers
985 views

How to Draw Complex Line Bundles

I am giving a presentation soon on the Classification of Complex Line Bundles and I would like to have some very "basic" visualizations to use as examples. Background and Context I am considering ...
6
votes
0answers
197 views

When is the cohomology cross product square nonzero?

Let $X$ be a finite CW complex, and $A$ an abelian group. Given a nonzero class in singular cohomology $$x\in H^k(X;A),$$ when is its cross product square $$x\times x\in H^{2k}(X\times X;A\otimes A)$$ ...
1
vote
1answer
215 views

Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative? In other words, is the following statement true? If it is true, then, how can one prove it? ...
2
votes
0answers
343 views

The cohomology of the relative dualizing sheaf of a relative curve

Let $X\to S$ be a curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$. I know that ...
3
votes
1answer
863 views

injectivity of the pull-back via a finite map

Let $f:X \to Y$ be a finite map of (smooth, compact) complex algebraic varieties. Then a map $f^*$ is defined at the level of Chow and cohomology rings. Say that for simplicity we work with rational ...
11
votes
1answer
1k views

Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1

So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$. For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney ...
3
votes
1answer
223 views

What is known about formality of flag varieties?

Let $G$ be a connected, complex reductive algebraic group and $X=G/P$ a (partial) flag variety. For example by "Real homotopy theory of Kähler manifolds", we know that its cohomology with real ...
14
votes
5answers
2k views

de Rham vs Dolbeault Cohomology

For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. ...
3
votes
1answer
267 views

Reference for the converse of Cartan's Theorem B

Cartan's theorem B states that every coherent analytic sheaf on a Stein manifold is acyclic. Hartshorne claims that the converse to this theorem also holds, but doesn't supply a reference. Does ...
4
votes
1answer
181 views

determinant of periods

Let $X$ be a projective, smooth algebraic variety defined over the field of algebraic numbers. Consider algebraic de Rham cohomology $H_{dR}(X)$ and singular cohomology of $X(\mathbb{C})$ with ...
0
votes
0answers
226 views

Kunneth formula (finite group cohomology) for non trivial action of group

Is there a Kunneth formula relating $H^i(k[G],A)\otimes_k H^i(k[G],B)$ and $H^i(k[G],A\otimes_k B)$ where $A\otimes_k B$ is given the diagonal $G$ action ?
8
votes
4answers
1k views

Program for computing group cohomology

Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space? I mainly care about infinite groups.
4
votes
0answers
227 views

Cech cohomology and set theory

Can anyone point me to places in the literature where modern set theory has been applied to say something about the Cech cohomology of connected non-metrizable compacta? I'm looking for something ...
1
vote
1answer
168 views

How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...
3
votes
1answer
599 views

The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base

Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. ...
5
votes
4answers
385 views

(Co)homological characterization of homotopy pullbacks

For a commutative square of spaces (of manifolds, or of simplicial sets): $$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & ...
1
vote
1answer
212 views

Criteria for acyclicity

Let $X$ be a smooth projective variety. Let $E$ be a line bundle (or, more generally, a vetor bundle) on $X$. Are there any nice criteria for acyclicity of $E$ (that is, for the property $H^i(X,E)=0$ ...
1
vote
0answers
200 views

What can one say about a smooth variety whose lower cohomology is trivial?

Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such ...
2
votes
1answer
568 views

Notions related to De Rham Cohomology

In R^2, we have the following abelian groups, some of which have R vector space structures, or even C vector space structures. Closed forms/exact forms real parts of analytic functions/harmonic ...
2
votes
2answers
241 views

Kodaira type vanishing for an ample divisor + effective divisor with SNC support

Norimatsu's lemma says that on a smooth projective complex variety $X$ of dimension $n$, then we have $H^i(X,\mathcal O_X(-A-E))=0$ for $i < n$ when $A$ is an ample divisor and $E$ is a simple ...
3
votes
2answers
539 views

Stiefel-Whitney classes of a projective space bundle

Hi! Let $\gamma_1$ denote the twisted line bundle over $S^1$ and add a trivial $(2k-1)$-bundle $\mathbb{R}^{2k-1}$. Consider the projective bundle $P(\gamma_1 \oplus \mathbb{R}^{2k-1})$ over $S^1$. ...
2
votes
0answers
198 views

Cohomology of a projective limit

If we consider an oriented locally finite graphs $X$, and we suppose that $X$ is the projective limit of a familly of oriented locally finite graphs : ...
6
votes
1answer
320 views

Continuous and smooth Lie groupoid cohomology

In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential ...
1
vote
1answer
289 views

Coboundary map on the cochain complex of abelian cosimplicial groups?

Maybe I'm looking at the wrong places, but I can't find a definition of the coboundary map on the cochain complex of abelian cosimplicial groups. What I have in mind is something similar to the ...
9
votes
1answer
558 views

The coproduct on the cohomology of a Hopf algebra

If $H$ is a Hopf algebra over a field $k$, the Hopf algebra cohomology $\mathrm{Ext}_H(k,k)$ is —as usual— an algebra, but the Hopfness of $H$ turns it into a Hopf algebra. Is there a reference ...
2
votes
1answer
244 views

Do cohomologically trivial line bundles affect morphisms?

Assume we have a locally free sheaf $R$ of associative $O_S$-algebras of rank $r^2$, possibly noncommutative, where $S=\mathbb{P}^2$ over $\mathbb{C}$. Furthermore given a locally free $O_S$-module ...
1
vote
1answer
355 views

vanishing of cohomology sheaves with supports and values in the multiplicative group

Let $X$ be a locally noetherian regular scheme and $Y$ be a closed subscheme of codimension $d > 0$ in every point. Why does it "immédiatement" (Grothendieck, Groupe de Brauer III, §6, p. 133 f.) ...
3
votes
0answers
189 views

limit of étale cohomology with supports

I would like to know if the following generalisation of Milne Étale Cohomology, Lemma III.1.16, p. 88 holds: (all cohomology groups with respect to the étale topology) Let $Y \hookrightarrow X$ ...
3
votes
0answers
154 views

Branched Covers of a Two-sphere with Infinite Degree

I would like to ask the following question: Let us consider the following branched cover $X_n$ of $\mathbb{P}^1$ that can be thought as a hyper-surface of $\mathbb{P}^1 \times \mathbb{P}^1$: ...
1
vote
1answer
805 views

Penner's formula for volume of the Moduli Space

In his paper "Weil-Petersson Volumes" Penner gives the following formula for the integral of a top-dimensional cohomology class $\omega$ on the moduli space $\mathcal M_g^s$ of $s$-punctured riemann ...
1
vote
1answer
178 views

combination of two duality theorems

For a variety $X$ over a finite field $k$, one combines étale Poincaré duality for $X \times_k \bar{k}$ with duality for $k$ ($H^0(k,M) \times H^1(k,M^\vee) \to H^1(k,\mathbf{Z}/n) = \mathbf{Z}/n$) to ...
6
votes
0answers
355 views

Cohomology of a space with coefficients in a Lie group?

The following is a very naive construction, and I am almost embarrassed to ask questions about it. Suppose I have a connected simple graph $\Gamma = (\Gamma_0,\Gamma_1)$ (i.e., vertex set, edge set) ...
4
votes
1answer
399 views

Formal Group Laws on Ring Spectra?

Given a (graded) ring $R$, to define a formal group law it is equivalent to define a ring homomorphism $\phi:L\to R$ where $L$ is Lazard's ring. Is there any notion of defining a formal group law on ...