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3 votes
0 answers
227 views

Is it possible to use the Cech complex to compute coherent cohomology in practice?

Suppose I have a closed subvariety $V$ of $\mathbb P^n \times \mathbb A^m$ given by explicit equations. Is it possible in practice to compute the coherent cohomology of $V$ with coefficients in a line ...
2 votes
0 answers
141 views

Computing the coherent cohomology of a quasiprojective variety

I have a quasiprojective variety given by some explicit quations. How do I compute its coherent cohomology (with coefficients in the structure sheaf)? Do I use the Cech complex for an open affine ...
8 votes
0 answers
644 views

Trying to understand "Shtukas"

I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
4 votes
1 answer
333 views

Reference request: category of sheaves of O-modules with coherent cohomology

Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of $\mathcal{O}_X$-...
3 votes
1 answer
393 views

Cohomological base change

$\require{AMScd}$ Consider the Cartesian diagram of Noetherian schemes and commutative rings $R$, $R'$: \begin{CD} N' @>{h'}>> N\\@VV{g'}V @VVgV\\ M' @>h>> M \\ @VVV @VVV \\ \mathbf{...
6 votes
1 answer
911 views

English translation of Borel-Serre, Le théorème de Riemann-Roch?

Would be happy to receive a translation in English of Borel and Serre's Le théorème de Riemann-Roch, Bulletin de la Société Mathématique de France, Tome 86 (1958) pp. 97-136, doi:10.24033/bsmf.1500, ...
6 votes
1 answer
334 views

Is $h^{0,k}$ a topological invariant?

Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$ such that $X(\mathbb{C})$ is homeomorphic to $Y(\mathbb{C})$. Is it true that $\dim_{\mathbb{C}} H^k(X,\mathcal{O}_X)=\dim_{\mathbb{...
6 votes
3 answers
1k views

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 ...
1 vote
0 answers
137 views

Quasicoherent analogue of a theorem on fiberwise acyclicity for etale cohomology

I am interested in knowing what (if any) is the quasicoherent analogue of the following result that I have paraphrased from SGA 4, exposé xv, Théorème 1.15: Let $g \colon X \to ...
7 votes
2 answers
1k views

Can one prove vanishing of higher direct images fiber-wise?

Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence. are the following statements equivalent? The derived direct image of $O_X$ is $O_Y$. ...
1 vote
1 answer
320 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 $H^i(X,E)=0$ ...