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This is a rewrite of a deleted question. I've decided to focus on one particular example mentioned in that question. Below a point means a closed point. Let $U$ be the set of irreducible non-cuspidal …

One can compute the Betti numbers of a smooth complete intersection $X$ of multidegree $d=(d_1,\dotsc,d_r)$ by induction on $r$. The case $r=0$ corresponds to $\mathbb{P}^n(\mathbb{C})$. A smooth comp …

Complementing other answers in this thread: First, when $n>0$, there is the Penner decomposition (see e.g. Harer, The cohomology of the moduli space of curves, LNM 1337 or Penner, Comm Math Phys 113, …

Let $H_{P,n}$ be the Hilbert scheme of subschemes of $\mathbb{P}^n(\mathbb{C})$ with Hilbert polynomial $P\in\mathbb{Q}[t]$, and let $U_{P,n}\to H_{P,n}$ be the flat universal family. Are there $n,P$ …

I personally won't recommend Bredon's book, rather Iversen's "Cohomology of sheaves" (especially if you are interested in the topological aspects/applications of sheaf theory). There is also Dimca's " …

Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be inte …

This does follow from GAGA via the spectral sequences associated to the dumb filtrations on the algebraic and analytic de Rham complexes of sheaves, see p. 96 of tome 29 of PMIHES in a paper of Grothe …

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" quest …

A mixed Hodge structure (mHs) on a commutative differential graded algebra (cgda) over $\mathbf{Q}$ is a mixed Hodge structure on the underlying vector space such that the product and the differential …

Many of the amazing results by Italian geometers of the second half of the 19th and the first half of the 20th century were initially given heuristic explanations rather than rigorous proofs by their …

In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with fiber …

This question is inspired by Relation between Hecke Operator and Hecke Algebra I remember having heard of yet another way of looking at Hecke operators acting on the spaces of modular forms for class …

Since my previous question Hyperelliptic loci in Teichmueller spaces resulted in two quick and helpful replies, let me ask another question in a similar vein: A smooth compact complex curve is call …

Suppose $K\subset \mathbb{C}^n$ is a compact subset and $f:\mathbb{C}^n\setminus K\to \mathbb{C}$ is a holomorphic function. Then, provided $n>1$, $f$ extends to a holomorphic function defined on the …

The length of this question has got a little bit out of hand. I apologize. Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe period …