Questions tagged [complex-analytic-spaces]
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11 questions
5 votes
0 answers
150 views
Is “being a universally closed map” a property preserved by analytification?
$\def\C{\mathbb{C}} \def\an{\mathrm{an}}$Let $X, Y$ be $\C$-schemes that are locally of finite type. Let $f:X\to Y$ be a morphism of $\C$-schemes, and denote $f^\an:X^\an\to Y^\an$ to the ...
1 vote
1 answer
330 views
When is the pullback of a coherent analytic sheaf again coherent?
Let $f\colon X\to Y$ be a morphism of complex analytic spaces (though I'm very happy to restrict to complex manifolds). Theorem (Grauert). The pushforward $f_*\colon\mathcal{O}_X\text{-mod}\to\mathcal{...
- 1,312
2 votes
0 answers
112 views
Inclusion in Hardy-Smirnov spaces for the analytic continuation of a Cauchy-Type integral with a continuous boundary function
Let $D$ be a bounded simply connected domain in the complex plane $\mathbb{C}=\{z=x+iy\}$ with a Jordan rectifiable boundary $C=\partial D$. Let $P_1$ and $P_2$ be two distinct points on $C$, and let ...
- 41
4 votes
0 answers
197 views
Universal derivations in complex analytic geometry
In algebraic geometry, we are used to the following universal property of K"ahler differential forms: Let $f: X \rightarrow S$ be a morphism of (Noetherian) schemes. Then there is a relative ...
- 131
1 vote
0 answers
172 views
Extending Grothendieck topology from analytic manifolds to spaces?
Let $k\text{-Man}$ denote the Euclidean site of $k$-analytic manifolds where $k=\mathbb{R}, \mathbb{C}$. In words, $k\text{-Man}$ is the usual category of real/complex analytic manifolds considered ...
- 3,060
2 votes
0 answers
71 views
Complex analytic descent along G-actions
Let $G$ be a complex Lie group acting on a complex analytic space $X.$ To be clear, I don't require $X$ to be reduced. Let $f: Y\rightarrow X$ be a smooth morphism such that the $G$-action lifts to $...
- 3,060
1 vote
0 answers
148 views
Explicit resolution of $\Omega^1_C$ for prestable curve $C$
Suppose $C$ is a complex projective curve (or a compact $1$-dimensional connected reduced complex space). If $C$ is smooth, then its module of differentials $\Omega^1_C$ is locally free. If $C$ is a ...
- 1,861
3 votes
1 answer
98 views
Bounded form in complex complete manifold
If $\alpha$ is a bounded form in a complex complete manifold $X$ (i.e $\sup_X|\alpha (x) |<\infty$, then $d\alpha$ is it also bounded? Rq: if $d\alpha$ is bounded then \alpha is not necessary ...
- 31
5 votes
1 answer
561 views
Schemes over (locally) ringed spaces: working over complex-analytic spaces, rigid-analytic spaces, formal schemes, etc
Monique Hakim developed in her doctoral thesis [1] the theory of relative schemes. These comprise, as a special case, the theory of schemes over (locally) ringed spaces. What makes the study of these ...
- 12.7k
6 votes
0 answers
236 views
Does there exist a notion of Chern classes in intersection cohomology?
First of all: I apologize for my mistakes, I'm a freshman in intersection cohomology. Let $X$ be a (compact) complex analytic space, let $L$ be a line bundle over $X$. Can one define a notion of ...
- 918
2 votes
1 answer
652 views
what's the cohomological dimension of a Stein space?
I want to know the "cohomological dimension" of a Stein space. I know that: for $X$ differential manifold and for every sheaf $F$ of abelian groups, I have $H_c^j(X,F)=H^j(X,F)=0$ for $j>...
- 21