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Results tagged with complex-geometry
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user 4144
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
14
votes
Accepted
Variety without a compactification whose complement is smooth
Presumably, you want $\bar X$ to be smooth as well. Then there are many examples. Here is a simple one. Let $\bar Y$ be smooth projective curve of positive genus. Now remove at least two points to get …
- 35.2k
6
votes
Accepted
Prefactor $2\pi i$ for Tate-Hodge structure
You are right that it is in some sense a matter of convention, but I claim it's a natural one.
Perhaps the easiest example to explain is $H=H_1(X)$, where $X=\mathbb{C}^*$. By Deligne, this carries a …
- 35.2k
7
votes
How does complex conjugation act on the Hodge filtration?
I thought it would be useful to give an explicit example to supplement Olivier Benoist's answer; I will use the same notation as in his answer. Let $f(x)\in \mathbb{R}[x]$ be cubic with distinct roo …
- 35.2k
3
votes
The indecomposable bundle on an elliptic curve
I'll answer your main question for $r=1$, but for any smooth projective variety $X$. Interestingly, the answer is implicit in another paper of Atiyah's Complex analytic connections in fibre bundles, w …
- 35.2k
5
votes
On Simpson's motivicity conjecture
In the 90's, Simpson proved that rigid local systems on a projective variety come from complex variations of Hodge structure, so it seemed a natural if bold leap to conjecture that they come from geo …
- 35.2k
3
votes
Accepted
Cohomology of singular curves
You don't need all this machinery in this case (unless your goal was to understand the machinery). You have to exact sequences
$$0\to W_0\to H^1(X) \to H^1(Y)\to 0$$
$$0 \to H^1(X')\to H^1(Y)\to \oplu …
- 35.2k
3
votes
Motivation behind spectral sequences
I suspect a previous comment of mine led to this question, so let me say a few words here. The basic problem is this: Suppose $(A^\bullet, F)$ is a filtered complex, then one wants to relate the (co …
- 35.2k
4
votes
Accepted
What does does the monodromy weight filtration represent?
Given a nilpotent endomorphism $N$ of a finite dimension vector space $V$, Jordan canonical form implies that we can decomponse $V$ into a sum of "blocks" on which we can find bases satisfying $Ne_1 = …
- 35.2k
4
votes
Accepted
Meaning of torsion points in a Roitman's theorem
It occurs to me that your question shouldn't be taken literally, and is simply asking about the meaning of the theorem. To appreciate it, one can ask what $CH_0(X)$ looks like. As a first attempt, ma …
- 35.2k
5
votes
Accepted
Relating the holomorphic Euler characteristic of a family of algebraic varieties to properti...
I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth p …
- 35.2k
8
votes
Accepted
Do we have the Oka coherence theorem for finite group actions?
This would be true. You need two facts:
Grauert's theorem that coherent sheaves are preserved by proper direct images. This implies
$\pi_*\mathcal{O}_{\mathbb{C}^n}$ is coherent.
Sub modules of cohe …
- 35.2k
6
votes
Gorenstein varieties: why the two definitions are equivalent?
I would say that definition of Gorenstein given in Hartshorne etc. is the correct one; it certainly doesn't require normality. For example, a singular plane curve is Gorenstein in this sense, but not …
- 35.2k
2
votes
Extension of closed $(1, 1)$-forms
Contrary to my earlier comments, I now believe (but I haven't completely checked) that this is false in general. (I prefer not to comment on the paper you linked.) Suppose that $X$ is a Kähler manifol …
- 35.2k
1
vote
Accepted
Period map for $\partial\bar\partial$-manifolds
Let me start with a disclaimer that I think the following facts are true, but I'm doing this over coffee and I haven't checked the details carefully. First, I'll redefine $F^pH^k(X,\mathbb{C})$ to be …
- 35.2k
7
votes
Accepted
Comparison of weight filtration on cohomology of complex manifold
Yes, the $\ell$-adic weight filtration is compatible with the weight filtration in mixed Hodge theory under the comparison isomorphism. These facts go back to Deligne, and are described in his announc …
- 35.2k