All Questions
Tagged with several-complex-variables complex-geometry
87 questions
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Currents with logarithmic poles compared with those with no poles
I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by
$$
'\...
- 161
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76
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Constant mean curvature hypersurface
Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if ...
- 37
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34
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Analytic continuation of a bi-holomorphic automorphism on an irreducible bounded symmetric domain
I need your help.
Let $\Omega \subseteq \mathbb{C}^n$ be a type $IV_n$ Cartan domain, i.e;
$\Omega$ =$\{ z \in \mathbb{C}^n$: $1-2Q(z,\bar z)+|Q(z, z)|^2>0,\qquad Q(z, \bar z)<1 \}$
where $Q(z,...
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0
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144
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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 ...
- 1,352
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39
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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
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1
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218
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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$, ...
- 1,170
3
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89
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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 ...
- 658
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88
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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\...
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1
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173
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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 ...
- 1,024
1
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86
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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 ...
- 9,019
2
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86
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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 ...
- 1,441
2
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1
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162
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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 $...
- 1,441
1
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198
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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.$$
Recall ...
- 295
4
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192
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Factorization of an analytic function in $\mathbb{C}^n$
Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z_1,\ldots,z_n]$ is a polynomial whose irreducible factors are all of multiplicity one.
If $...
- 4,034
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1
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461
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Determine the coefficient of the exceptional divisor
Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) ...
- 295
5
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1
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442
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Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?
The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...
- 1,141
3
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0
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98
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Analogous tensor product operation for reflexive sheaf
Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it.
Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
- 295
1
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1
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270
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Global sections of a line bundle on a reducible complex space
Let $S$ be a reducible compact complex analytic space, thus we have the decomposition $S=\bigcup_{i=1}^n {V_i}$ where $V_i$ is the irreducible component of $S$. Let $L$ be a line bundle on $S$, I ...
- 295
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132
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On the definition of Cauchy transform [closed]
I have seen two different definitions of the Cauchy transform of a smooth function one is with respect to the line integral (for eg. in. the book "The Cauchy transform and potential theory")...
- 21
2
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1
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238
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Extension of a Szegő Kernel to the boundary
Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.
Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...
- 63
3
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177
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A question on the proof of Bedford-Taylor theorem in Demailly's book
I am trying to understand a proof of the Bedford-Taylor theorem on the weak convergence of Monge-Ampere operators of decreasing sequences of plurisubharmonic functions.
I am reading a proof in the ...
- 21.8k
1
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271
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Starlike sets in $\mathbb{C}^n$
Let $S$ be a bounded domain in $\mathbb{C}^n$. $S$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. If $S$ ...
- 61
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204
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Do we have the Oka coherence theorem for finite group actions?
We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By Oka coherence theorem, $\mathcal{O}(\mathbb{C}^n)$ is coherent over itself.
Now we consider a ...
- 9,019
7
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1
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403
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Analogue of Grauert's upper semi-continuity for Bott–Chern cohomology
In Coherent analytic sheaves, one has the following theorem due to Grauert:
Let $f: X \rightarrow Y$ be a holomorphic family of compact complex manifolds with connected complex manifolds $X, Y$ and $V$...
- 295
1
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0
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88
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Holomorphic mapping on a manifold approximating a constant map
Let $X,Y$ be complex manifold, $Y$ Stein. It sounds quite reasonable to formulate the following claim: given $y_0\in Y$, for every $\epsilon>0$ and $M\subset X$ compact, there exists an holomorphic ...
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77
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On extension of Monge-Ampere masses
It is known (evidently due to Bedford and Taylor) that if $u$ and $v$ are bounded plurisub-harmonic functions on an open domain in $\mathbb{C}^n$, then $$\mathbb{1}_{\{u>v\}}(dd^cu)^n = \mathbb{1}_{...
- 537
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0
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70
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Regular exposable points on the boundary of compacts in Stein manifolds
Given a Stein manifold $Y$, there exists $\rho$, a $\mathscr C^2$-smooth strictly plurisubharmonic exhausting function for $Y$, such that the set of critical points $C=\{z\in Y\;:\;d\rho(z)=0\}$ is ...
- 779
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138
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Are open subsets of a $\sigma$-compact LCH space $\mathcal{K}$-analytic?
I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\...
- 1,141
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1
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351
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Oka-Grauert principle, up to the boundary
Let $Z\subset \mathbb{C}^n$ a domain of holomorphy with smooth boundary $\partial Z$ and closure $\bar Z$. There is a natural notion of holomorphic vector bundle over $\bar Z$, given in terms of ...
- 779
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217
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Is any proper subvariety contained in hypersurface
Suppose $A$ is a subvariety of an irreducible complex space(analytic variety) $X$. Is there an analytic hypersurface of $X$ containing $A$?
- 11
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120
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How to get the jet extension over the whole of $X$ in Popovici's article?
Recently, I am reading D. Popovici's article $L^2$ extension for jets of holomorphic sections of a Hermitian line bundle, https://arxiv.org/pdf/math/0409170.pdf where some parts possibly confuse me.
I ...
- 295
2
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0
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120
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A complex analytic interpretation of multiplicity on the special fiber of a flat family
Let $X$ be a variety over $\mathbb C$ and $\pi: X\to \Delta$ be a flat morphism over the unit disk $\Delta=\{z:|z|<1\}$. Let $Z$ be a component of $X_0=\pi^{-1}(0)$. The multiplicity of $Z$ is ...
- 1,803
4
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1
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317
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Existence of plurisubharmonic functions on complex manifolds
Let $X$ be a noncompact complex manifold which contains no positive dimensional compact analytic sets.
Conjecture: There must be strictly plurisubharmonic functions on $X$ .
Is it true?
- 391
5
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1
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288
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Non-constant holomorphic map onto a smooth curve
Let $\Gamma$ be a smooth projective curve in $\mathbb{P}^2$ and let $U$ be an open neighborhood of $\Gamma$. Denote by $\Gamma_1,\Gamma_2,\ldots,\Gamma_n$ a finite collection of smooth curves ...
- 293
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0
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116
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How to regard negative PSH function with neat analytic singularities as a generalization of Green-type function?
I am reading this paper:A SIMPLIFIED PROOF OF OPTIMAL L2-EXTENSION THEOREM AND EXTENSIONS FROM NON-REDUCED SUBVARIETIES by Hosono. https://arxiv.org/pdf/1910.05782.pdf.
The setting is as follows.Let $...
- 295
3
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1
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194
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Proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$
I am a PhD student in several complex variables.
I am reading this paper by Orevkov proving that there exists a proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$.
I ...
- 779
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52
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Constructing certain Global section with prescribed zero locus over Stein manifold
Let $X^n$ be a Stein manifold (complex submanifold in $\mathbb{C}^N$ for some large $N$). Let $D = \{(z,z)\in X\times X: z\in X\}$ be the diagonal in $X\times X$. I'm looking for some holomorphic ...
- 111
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212
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$(-2)$-curves in complex $3$-folds
Let $X$ be a smooth complex $3$-fold,
and let $C \subset X$ be an embedded smooth rational curve whose
normal bundle $N_{C/X}$ is isomorphic to $\mathscr{O}(-1) \oplus \mathscr{O}(-1)$.
Is it true ...
- 81
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3
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930
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Complex manifold with boundary
My question is of local nature.
Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.
Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)...
- 43.2k
2
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0
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251
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Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?
Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...
- 9,019
2
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0
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91
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Automatic plurisubharmonicity for a non-negative function
I feel confused about a point in this very short paper. On the top of page 3, it is claimed that:
If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function ...
- 2,789
1
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0
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40
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On Remmerts reduction
Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...
- 1,409
1
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0
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181
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Interpretation of deformation of complex structure
Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq ...
- 11
6
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0
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241
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Bezout theorem for germs of holomorphic functions
UPDATE.
It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample.
Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
- 1,238
0
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1
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847
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Exponential Sequence of Sheaves
Let $(X, \mathcal{O}_X)$ be a complex analytic space in the sense of Grauert, i.e., a $\mathbb{C}$-analytic ringed space which is locally isomorphic to a local model. We may assume that $X$ is a ...
- 1,379
5
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0
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543
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a question on Hodge and Atiyah's paper "integrals of the second kind on an algebraic variety"
I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71:
$$H_{2n-q}(S)...
- 1,121
17
votes
2
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2k
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Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology
While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically,
When defining Dolbeault ...
- 419
2
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0
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75
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Notation and geometry facts in a paper on the Diederich-Fornæss index
I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.
I am having some problems with both the notation and the geometrical side.
1)I don't know what kind of objects $N,L$ are ...
- 779
0
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1
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229
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Can any plurisubharmonic function be represented as a sum of non-positive plurisubharmonic functions?
Let $f$ be a plurisubharmonic function, $f < 0$ in $\Omega$. Can we always find two negative plurisubharmonic functions $u$ and $v$ and real numbers $a,b\in(-1,1)$ such that
$$-f=(-u)^{a}+(-v)^{b}$...
3
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0
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193
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What do we necessarily need for the image of a domain of holomorphy to be a domain of holomorphy
I posted this on Math.Stack.Exchange with no luck, so I thought it would be perhaps better suited for this site.
We recall that a domain of holomorphy is a domain in $\mathbb{C}^n$ that is ...
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