# Questions tagged [sheaf-cohomology]

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. (Def: http://en.m.wikipedia.org/wiki/Sheaf_cohomology)

806 questions
Filter by
Sorted by
Tagged with
34 views

### Project of understanding proof that the ampleness of pull back implies ampleness under some condition ( Gortz's Algebraic Geometry book, Vol.2. )

I am reading the Gortz's Algebraic Geometry, Vol.2, proof of Lemma 23.7 and stuck at some argument Lemma 23.7. Let $X$ be a quasi-compact ( quasi-separated ) scheme, and let $i: X' \to X$ be a closed ...
• 2,415
1 vote
34 views

• 1,200
77 views

### How to show naturality of the section when trying to prove gluing axiom for sheaf hom

The question is to show that the sheaf Hom$(U):=\{$morphisms of sheaves $\mathcal{F}|_{U}\to \mathcal{G}|_{U}\}$ is indeed a sheaf, where $\mathcal{F}$ and $\mathcal{G}$ are fixed sheaves on the same ...
• 197
96 views

### Cohomology of twisted tangent bundle of fake projective plane

Let $X$ be a fake projective plane, id est $X$ is a smooth projective surface with the same Betti numbers of $\mathbb{P}^2_{\mathbb{C}}$ but not isomorphic to $\mathbb{P}^2_{\mathbb{C}}$. These ...
• 3,907
39 views

• 439
93 views

### Lazarsfeld's proof of Mumford's regularity Theorem 1.8.3

I want to apologize first of all if my English is bad, I hope my message will still be understandable. I’m currently reading Lazarsfeld’s book "Positivity in Algebraic Geometry I" and I’m ...
• 529
52 views

### Hodge decomposition metric dependency

I’m currently studying Complex Geometry by Daniel Huybrechts and I can’t understand why it is crucial to prove that Hodge Decomposition does not depend on the chosen metric. I’m talking about ...
42 views

### Direct sum of two sheaves

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two sheaves on $X$ ,is $H^{i}(X,\mathcal{F}_1 \oplus \mathcal{F}_2) \simeq H^{i}(X,\mathcal{F}_1) \oplus H^{i}(X,\mathcal{F}_2)$ ?
92 views

### Sheaf cohomology on quotient stacks

Suppose I have a scheme $X$ over $\mathbb C$, acted on by a finite group $G$. Let $\mathcal F$ be a $G$-equivariant coherent sheaf of $\mathcal O_X$-modules. Then I can form the stack quotient $[X/G]$,...
1 vote
65 views

• 602
124 views

### ADHM construction: why two bundles of the monad have the same ranks?

I'm reading The ADHM construction of Yang-Mills instantons by Simon Donaldson. A theorem in the section 4 says: Let $E$ be a rank $r$ holomorphic bundle over $\mathbb{CP}^3$ with $c_2 = k$. Suppose ...
89 views

### What is "torsion" in the context of cohomology, and why is it important?

I searched for some answers, but most answers discussed the meaning of torsion, instead of its definition. Not knowing how the torsion is defined (in cohomology) I couldn't understand those answers at ...
Consider $X$ smooth projective curve such that $X\subseteq \mathbb P^3$, $X$ is not contained in any hyperplane, and $g(X)=1$. In this exercise the idea is to consider the long exact sequence we get ...
### Is left t-exactness of $Rf_\ast$ for $f : X\longrightarrow Y$ affine just Andreotti-Frenkel
Sorry I'm a little lost in those perverse sheaves. I think this left t-exactness property is called Artin-Grothendieck vanishing, which I take to be the relative version of $H^n(X,\mathcal{P})=0$ for \$...