All Questions
Tagged with complex-geometry cohomology
53
questions
2
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0
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132
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Hodge numbers of a complement
Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that
$$
H^i(X) \to H^i(...
- 2,305
2
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0
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177
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Splitting of de Rham cohomology for singular spaces
I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
- 513
11
votes
1
answer
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Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
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2
votes
1
answer
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A specific question on the Griffiths' paper: the reduction of the pole order
If someone has gone through the Griffiths' paper ``On the periods of certain rational integrals: I,'' could you help me to understand Lemma 8.10?
I don't get why $\eta\in Z^{q,k+1}(l-1)$; although $\...
- 63
4
votes
0
answers
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Does the reduction of the pole order to compute the Poincare residue work?
I am trying to understand the Poincare residue and referring to On Computing Picard-Fuchs Equations, which is cited by Wikipedia's page on the Poincare residue.
On pp. 5--6, he gives a way to compute ...
- 63
5
votes
1
answer
323
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Top integer homology of compact analytic variety
Let $V$ be a compact connected complex analytic subvariety (possibly singular) of a complex smooth manifold. Let $n$ denote its complex dimension.
Is it true that $H_{2n}(V,\mathbb{Z})\simeq \mathbb{Z}...
- 21.3k
1
vote
0
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110
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Cohomology of the base of an elliptic fibre space
Work over $\mathbb{C}$. Let $\Phi : X \to S$ be an elliptic fiber space, where $X$ is a smooth projective threefold with $H^1(\mathcal{O}_X)=H^2(\mathcal{O}_X)=0$, and $S$ is a smooth projective ...
- 1,369
1
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0
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79
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Representatives of line bundle cohomology over tori
Let $V^n$ a be a $\mathbb{C}$-vector space. For $U\subset V$ a complete lattice, the holomorphic line bundles over $V/U$ are classified (see e.g. `Abelian varieties', D. Mumford) by data $(H,\alpha)$ ...
6
votes
0
answers
511
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What is the geometric meaning of $H^2(X, \mathscr{O}_X)$?
Let $X$ be a compact complex manifold with structure sheaf $\mathscr{O}_X$ (the sheaf of holomorphic functions on $X$).
What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$?
In the ...
- 1,369
3
votes
1
answer
648
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Cohomology of singular projective cubic surface
Let $X\subset \mathbb{P}_{\mathbb{C}}^3$ be a projective singular cubic surface with two singular points. Is the rationalcohomology of such objects known? As an example of the type of surfaces I'd be ...
6
votes
0
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214
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Does the stable homotopy type of a variety depend on an embedding into C?
Crossposting from math SE since my question might not be as well-known as I had assumed.
Suppose I have an algebraically closed field $K$ of finite transcendence degree over $\mathbb{Q}$. To any ...
- 341
6
votes
1
answer
552
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How do I remember which power of the Lefschetz operator $L$ corresponds to the $k$th Primitive cohomology group?
Let $X$ be a compact Kähler manifold with $L$ denoting the Lefschetz operator $L(\bullet) = \bullet \wedge \omega$. The primitive cohomology groups are defined, for $k \in \mathbb{N}$, by $$P^k(X, \...
- 651
4
votes
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294
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Holomorphic covers pulling back the volume form to any integer multiple
Let $M$ be a closed connected complex manifold with $\mathrm{dim}\:M=n$. Can there exist holomorphic covering maps $\phi_k:M\to M$ for all integers $k\geq 1$ such that $\phi_k^*:H^n(M, \mathbb{Z})\to ...
user145520
1
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0
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203
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Find torsion classes using flat bundles
My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arose a discussion when the map $H^2(X,\mathbb{Z}...
- 5,780
1
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0
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115
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3rd Cohomology of a fibration with Flag varieties as fibers
Let $X$ be a smooth projective rational variety over $\mathbb{C}$, let $Y$ be another smooth projective variety, both of dimension bigger than 2, and let $\pi : Y \rightarrow X$ be a locally trivial ...
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