All Questions
Tagged with complex-geometry several-complex-variables
85
questions
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126
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Function of several complex variables with prescribed zeros
I accidentally stumbled upon a problem of complex analysis in several variables, and I have a hard time understanding what I read, it might be related to the Cousin II problem but I cannot say for ...
- 1,332
0
votes
0
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33
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Contraction of an inclusion with respect to Kobayshi hyperbolic metric
Suppose that $X = \mathbb{C}^n - \Delta_X$ and $Y = \mathbb{C}^n - \Delta_Y$, where $\Delta_X$ and $\Delta_Y$ are unions of hyperplanes in $\mathbb{C}^n$ such that $\Delta_Y \subset \Delta_X$, $\...
- 41
3
votes
1
answer
199
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Subset of a complex manifold whose intersection with every holomorphic curve is analytic
The Osgood's lemma in complex analysis says that if $f \colon \mathbb{C}^n \to \mathbb{C}$ a continuous function such that $f|_{L}$ is holomorphic for every complex line $L \subset \mathbb{C}^n$, ...
- 1,160
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78
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Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?
Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
- 538
1
vote
0
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58
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Complex geodesic coordinate, local ramified map, and the conic metric
Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance
Let $X$ be a compact Kaehler ...
- 295
2
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answers
85
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Does Kobayashi isometry map preserve complex geodesics?
Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...
3
votes
1
answer
156
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$n$-th root of meromorphic functions of several complex variables
Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true.
Claim. $f$ admits a global ...
- 1,014
1
vote
0
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79
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Do we have a Grauert-Fischer theorem for non-trivial families?
This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a ...
- 8,767
2
votes
0
answers
83
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Abelian subgroup of the automorphism group of $\mathbb C^n$
Let $Aut(\mathbb C^n)$ be the automorphism group of $\mathbb C^n$, i.e., the group of all biholomorphic maps of $\mathbb C^n \to \mathbb C^n$. Suppose $T$ is a finitely dimensional torus which is a ...
- 1,401
2
votes
1
answer
128
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1-convex and holomorphically convex
A complex manifold $M$ is called $1$-convex if there exists a smooth, exhaustive, plurisubharmonic function that is strictly plurisubharmonic outside a compact set of $M$.
Can we prove that if $M$ is $...
- 1,401
1
vote
0
answers
186
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Determinant of the conormal bundle
Let $Y$ be a smooth submanifold of codimension $r$ in a complex manifold $X$. By virtue of the adjunction formula, we always have the isomorphism
$$K_Y\simeq (K_X\otimes \det N_Y){\,|\,}_Y.$$
Recall ...
- 295
4
votes
1
answer
173
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Factorization of an analytic function in $\mathbb{C}^n$
Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z_1,\ldots,z_n]$ is a polynomial whose irreducible factors are all of multiplicity one.
If $...
- 3,924
1
vote
1
answer
446
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Determine the coefficient of the exceptional divisor
Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) ...
- 295
4
votes
1
answer
424
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Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?
The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...
- 1,111
3
votes
0
answers
93
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Analogous tensor product operation for reflexive sheaf
Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it.
Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
- 295