5
votes
1answer
83 views

Bounding the cohomology of a smooth projective variety

Let $X/\mathbb C$ be a smooth projective variety. Suppose it is smoothly embedded in $\mathbf P^n$ as the zero locus of an ideal generated by homogeneous polynomials $f_1, f_2, \dots, f_r$ in $n+1$ ...
2
votes
2answers
85 views

Abelian category without enough injectives

What is an example of an abelian category that does not have enough injectives? An example must exist, but I haven't been able to find one. If possible, a brief explanation of why the abelian ...
2
votes
0answers
59 views

cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
4
votes
0answers
71 views

Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have ...
6
votes
0answers
233 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
1
vote
1answer
37 views

Quantum Cohomology of Affine Toric Varieties

I would like to know whether quantum cohomology rings of affine toric varieties have been calculated, if this is possible. Does anyone have a relevant paper they could refer me to? I have seen it ...
4
votes
0answers
64 views

Geometric Interepretation of $\mathbb{G}_a$-torsors

Let's fixed a locally ringed space $(X,\mathcal{O}_X)$ (although, this should apply to any ringed topos, but I haven't thought that through). In fact, if it's helpful, you can assume that $X$ is a ...
3
votes
1answer
67 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
1
vote
0answers
44 views

Tensor product with an L on top

Looking at the definition of the Kunneth morphism in SGA4, XVII, 5.4.1.4, there is the notation $$Rf_*K \overset{\mathbb{L}}{\boxtimes}_{\mathcal{A}_0} Rg_* L \rightarrow Rh_*(K ...
0
votes
0answers
36 views

Book Suggestion - Complex algebraic surfaces

I am studying for an exam of algebraic geometry, in particular, I am dealing with ruled surfaces and numerical invariants, rational surfaces, Castelnuovo's Theorem and its application. I am reading ...
0
votes
0answers
45 views

Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
4
votes
0answers
34 views

Are acyclic coverings cofinal in the set of coverings?

I am interested by the following question in algebraic geometry. Recall that a covering $\mathfrak{U}$ of a topological space $X$ is acyclic for a sheaf $\mathscr{F}$ if we have $H^q(U_{i_0,\cdots, ...
3
votes
0answers
51 views

For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
3
votes
0answers
53 views

The vanishing (?) cohomology of the Milnor fiber

Setup. Say we have a germ of a holomorphic function $f:(\mathbb C^{n+1},0)\to (\mathbb C,0)$ with a critical point at the origin. There is an $\epsilon>0$ small enough so that $f$ becomes a ...
4
votes
1answer
61 views

Can the cohomology vanish on $X$ but on no restriction to hyperplane sections?

I would need a result asserting that if, say, a locally free sheaf $\mathcal{F}$ on a projective $X$ has non-vanishing cohomology when restricted to any smooth hyperplane section, then $H^1(X, ...
1
vote
1answer
34 views

Vanishing of local cohomology groups

Let $k$ be a field and let $X$ be a smooth separated $k$-variety. Let $T$ be a closed integral subscheme of $X$ of generic point $\eta$. The object of interest here is the local cohomology group $$ ...
6
votes
1answer
58 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
5
votes
1answer
80 views

Cohomology of the structure sheaf of $\mathbb{P}^1 \times \mathbb{P}^1$

I need to compute the euler characteristic $\chi(\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1})$ of the structure sheaf of $\mathbb{P}^1 \times \mathbb{P}^1$ (the product of the projective complex ...
1
vote
0answers
31 views

Computing $H^k(\mathbb{C}P^n \times \mathbb{C}P^m, \mathcal{O}^*(\mathbb{C}P^n \times \mathbb{C}P^m))$.

I have to calculate Cech cohomology groups $H^k(\mathbb{C}P^n \times \mathbb{C}P^m, \mathcal{O}^*(\mathbb{C}P^n \times \mathbb{C}P^m))$, working in Cech cohomology, where $\mathcal{O}^*$ is the sheaf ...
1
vote
1answer
43 views

Why do we have $H_2(X,\mathbb Z)\cong\mathbb Z$ for the quintic threefold $X\subset\mathbb P^4$?

Let us work over $\mathbb C$. In this article by S. Katz, it is stated that for a quintic threefold $X\subset \mathbb P^4$ one has $$H_2(X,\mathbb Z)\cong\mathbb Z.$$ Can anyone help me to see why ...
2
votes
0answers
41 views

$H^k(\mathbb{C}P^2 \times \mathbb{C}P^2, \mathcal{O}^*(\mathbb{C}P^2 \times \mathbb{C}P^2))$

I'd like to try to compute Cech cohomology groups $H^k(\mathbb{C}P^2 \times \mathbb{C}P^2, \mathcal{O}^*(\mathbb{C}P^2 \times \mathbb{C}P^2))$, but I don't know how can I do it. In my notes the author ...
2
votes
0answers
35 views

Algebraic Variety compact cohomology and singular cohomology.

Given an algebraic variety $X$ defined over the complex numbers and its compact cohomology $H^i_c(X)$, under what conditions it is possible to compute its singular cohomology $H^i(X)$.
3
votes
1answer
77 views

Vanishing of higher direct images of a composition

In a paper I am studying we have the following situation. Let $S$ be the spectrum of a Dedekind domain, and let $X$, $Y$ and $Z$ be scheme of finite type over $S$, where $X$ and $Y$ are smooths and ...
2
votes
1answer
83 views

Sheaf cohomology of $\mathbb{P}^3$

Let $\mathbb{P}$ denote the projective space over $\mathbb{C}$. In some lecture notes I found the claim that $$ h^0(\mathbb{P}^3, \mathcal{O}(2)) = 10 $$ Do you know why this is the case? In ...
3
votes
1answer
47 views

Global sections of a twisting of a structure sheaf of a projective scheme

Let $X$ be a projective Noetherian scheme over $\mathbb{C}$. Is it true that $H^0(\mathcal{O}_X(-t))=0$ for any $t>0$?
2
votes
1answer
64 views

Vanishing of $R^1f_*\mathcal O_X$

I am probably missing something obvious here, but none the less, here goes: Is the following statement (or perhaps some minor modification of it) true and if so, why: $R^1f_*\mathcal O_X = 0$ for a ...
4
votes
1answer
96 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
2
votes
0answers
70 views

Relation between algebraic hyper de Rham cohomology and hodge theory in positive characteristic

I have recently been looking at algebraic de Rham cohomology of curves in positive characteristic. In particular, I am looking at when the sequence $$0 \rightarrow H^0(X,\Omega_X) \rightarrow ...
1
vote
0answers
81 views

Derived push-forward of projective sheaf

Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves. Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding. ...
0
votes
0answers
47 views

When is $\mathcal{H}om$ functor exact in the category of presheaves

Let $C$ be a projective $\mathbb{C}$-scheme of pure dimension $1$. Suppose that $C$ is local complete intersection in $\mathbb{P}^3$. Let $C_1$ be an irreducible component of $C$, also of pure ...
5
votes
0answers
68 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
1
vote
0answers
27 views

Simple question on splitting of cohomology groups.

From the exponential exact sequence, I have $$ 0 \rightarrow H^2(X,\mathbb{C})/H^2(X,\mathbb{Z})\rightarrow H^2(X,\mathbb{C}^\times) \rightarrow Tor(H^3(X,\mathbb{Z})) \rightarrow 0. $$ for some ...
3
votes
1answer
96 views

Global section of pull-back of structure sheaf of projective scheme

Let $X$ be a smooth projective variety and $Z_1, Z_2$ two smooth projective divisors in $X$. Is it true that the natural restriction morphism from $H^0(\mathcal{O}_X(-Z_1-Z_2))$ to $H^0(\mathcal{O}_X ...
8
votes
0answers
100 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
8
votes
2answers
104 views

Cohomological definition of the Chow ring

Let $X$ be a smooth projective variety over a field $k$. One can define the Chow ring $A^\bullet(X)$ to be the free group generated by irreducible subvarieties, modulo rational equivalence. ...
7
votes
1answer
116 views

Strange case of Serre's duality

$\newcommand{\O}{\mathcal{O}}$ Let $X$ be a smooth projective curve and $D$ and effective divisor on it. The normal bundle of $D$ is defined as $$ \O_D(D)\; = \; \O_x(D)\;\otimes_{\O_X}\, \O_D$$ where ...
5
votes
2answers
74 views

Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.

Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ if $m < 0$. This statement came up in an algebraic geometry text with no explanation ...
18
votes
1answer
271 views

Why is there “no analogue of $2i\pi$ in $\mathbf C_p$”?

In his paper Fonctions L p-adiques, Pierre Colmez says: Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques ...
2
votes
0answers
61 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
4
votes
2answers
100 views

Cohomology and Global Sections

For a topological space X, $ \ H^0 (X, \Bbb Z)$ tells you about the connected components of $X$. For a sheaf $\mathcal O_X$ on $X$, $H^0 (X, \mathcal O_X)$ is usually written to refer to global ...
5
votes
0answers
97 views

Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
4
votes
1answer
57 views

Tensoring a flasque resolution by a line bundle give s a flasque resolution

I had an argument explained to me the other day and I didn't quite understand one of the steps. Here's my best reconstruction: Let $\mathscr{F}$ be a quasi-coherent sheaf, and $\mathscr{L}$ a ...
7
votes
1answer
196 views

A basic (?) question about Tate twists in étale cohomology

I have a basic question about the meaning of Tate twists in étale cohomology. I want the understand a statement of the form $$H^1(U,\Lambda) \cong \Lambda(-1)$$ in which $U$ is the spectrum of a ...
1
vote
0answers
54 views

A reference about Dolbeault cohomology

I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
0
votes
1answer
82 views

Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
4
votes
0answers
36 views

$H^{1}(O_{F})$ of a surface in a toric variety

I have a surface inside a toric variety $X$ and I would like to compute the first cohomology of its structure sheaf via the Cech complex, since I already know which cones of $X$ it hits (five ...
2
votes
0answers
129 views

Computing algebraic de-Rham cohomology via Čech cohomology

I have been reading this paper about de-Rham cohomology of hyperelliptic curves, and I have been trying to recompute some of what has been done in section 3. In particular, I am trying to see why ...
2
votes
0answers
99 views

Is there a (intuitive) connection between injective resolutions and $Hom( -, G)$ on a chain complex?

I understand the Idea of cohomology in the context of algebraic topology and sheaf cohomology. And I have a feeling about whats mesaured by the cohomology groups (in fact by the first). Further I see ...
17
votes
2answers
270 views

What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?

Let $X$ be a non-singular projective variety over $\mathbb{Q}$. Consider on the one hand $H^i_B(X(\mathbb{C}),\mathbb{Z}_\ell)$ the singular cohomology with value in $\mathbb{Z}_\ell$, and on the ...
1
vote
0answers
38 views

Dolbeault cohomology of $S^{2n-1} \times S^1$

Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}}(X)$ and $H^{(0,1)}_{\bar{\partial}}(X)$. I don't know how to do this but if we use Kunnet formula we have that ...