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Results tagged with complex-geometry
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user 49151
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.
4
votes
Accepted
Vector bundles over a Stein space are projective
I suppose you were given essentially a reference in the comment. Alternatively, to expand a little on the proof of this fact that I mention in the answer above the linked comment, to me this argument …
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4
votes
Accepted
Gluing local holomorphic connections
(Updated, as I realized the formula stated by Huybrechts indeed also does make sense without making any identifications.)
One should interpret the two sides in the compatibility condition of Huybrecht …
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3
votes
Accepted
Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)...
Yes, since this corresponds to a proper intersection, this kind of product is very robust and can be defined for a couple of different reasons:
Since the unbounded loci of $\log |z_1|^2$ and $\log |z …
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2
votes
Accepted
Resolving complexes of coherent analytic sheaves
If I interpret your question correctly, then I believe there is indeed such a construction.
The construction relies first of all on the existence of local resolutions as in your point 1. Secondly, it …
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8
votes
Accepted
How to define a current on a complex analytic space
The definition can be found in for example Section 3.3 of T. Bloom, M. Herrera: De Rham Cohomology of an Analytic Space (Inventiones math. 7, 275-296 (1969)) and Section 4.2 of M. Herrera, D. Lieberma …
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2
votes
Accepted
Do all closed positive currents lift to a resolution?
I believe the answer is no.
If I am reading the article below by Méo correctly, he shows that if
$\pi : Y \to X$ is the blowup of the polydisc $X=D^m$ along $Z := \{ z_1 = \dots = z_k = 0 \}$, and if …
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1
vote
What is the cohomology of the $C^{\infty}$-Koszul complex on $\mathbb{C}^2$?
As Vladimir mentions in the comment, it is probably easier to formulate this without the $\mathbb{C}$ at the right. However, it is not clear to me that the answer then is exactly what Vladimir claims, …
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1
vote
Equivalent definitions of normality for complex algebraic varieties
I don't know enough about the algebraic side to answer your question as stated, but the corresponding statement in the analytic setting is that the ring of weakly holomorphic functions on an irreducib …
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12
votes
Accepted
Holomorphic Urysohn Lemma
Yes, since the morphism of sheaves $\mathcal{O}_{\mathbb{C}^n} \to \mathcal{O}_{M \cup N}$ is surjective, it is surjective on global sections by Cartan's theorem B. Thus, any holomorphic function on $ …
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2
votes
Accepted
Characterization of a domain of holomorphy
I can't really anticipate what you will find ugly and hard to digest, but at least I found the proof in Jiří Lebls book "Tasty Bits of Several Complex Variables", Theorem 2.6.3, to be nicely presented …
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6
votes
Residues in several complex variables
I believe residue currents encompass most definitions of residues in several complex variables. Residue currents as developed in the 20th century are discussed for example in the survey "Residue curre …
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3
votes
Lelong numbers and integrability of psh functions
If you want $\alpha$ to not depend on more than $\nu(\varphi,0)$ and you want it to hold in $B_{1/2}(0)$, as I have interpreted your question now, then the answer is no:
In $\mathbb{C}$, you can take …
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5
votes
Chern classes and singular hermitian metrics on vector bundles
As Hassan mentions, in the setting of singular metrics on vector bundles, the notion of curvature appears problematic, which is discussed in the paper of Raufi.
Still, as is also discussed in that p …
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3
votes
Accepted
Continuations of holomorphic functions on submanifolds to the total space
The construction as suggested by Loïc Teyssier can be globalized, with the help of a Stein neighborhood basis.
By Corollary 1 in ''Every Stein subvariety admits a Stein neighborhood'' by Siu, every St …
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