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Results tagged with complex-geometry
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user 4696
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.
2
votes
1-convex and holomorphically convex
The answer to your question is yes , 1-convex implies holomorphic convexity. This is Grauert's solution of the Levi problem.You can find a proof in the book of Fritzsche and Grauert titled From Holomo …
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3
votes
Accepted
A cohomological variant of the second Riemann's extension theorem
For the first cohomology statement, you need the codimension to be at least three and for the first and second cohomology, codimension four. This theorem was proved by G Scheja in [1]. You can also fi …
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5
votes
Accepted
Dimension of intersection of real analytic sets
For a counterexample take a sphere in 3 space and a plane tangent to it .
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7
votes
Accepted
Connectedness of boundary of a Stein domain
This is a consequence of the fact that any Stein manifold with complex dimension at least 2 has one end. This follows from Hartogs extension across compact sets in Stein manifolds. You can find a proo …
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18
votes
Accepted
Hartogs' theorem for real-analytic subvarieties
The answer to your question is yes. It is enough to have the $2n-2$
dimensional Hausdorff measure of $X$ be zero and $X$ is closed. See the book of E. M. Chirka Complex Analytic Sets, page 298 proposi …
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27
votes
Accepted
Does every group arise as the fundamental group of a complete Kähler manifold?
Any Stein manifold admits a complete Kähler metric. Start with a connected real analytic manifold with the given fundamental group. A suitable tubular neighbourhood of the complexification will be St …
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3
votes
Accepted
Normal Cones for Complex Spaces
Proposition 1.17 in the paper Complex Analytic Cones by Axelsson and Magnusson in
Math Ann 273 pages 601-628 answers your question.
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4
votes
Does there exists a complete Kahler metric with positive scalar curvature on Poincare disk?
The answer to your question is no.In case of surfaces the scalar curvature and Ricci curvature coincide with Gauss curvature. If an n dimensional M had a complete Riemannian metric h with nonnegative …
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5
votes
Accepted
Monodromy representations and branched covers
For a modern treatment of the Grauert Remmert argument see Chapter 4 in the book Several Complex Variables vol 7 by Dethloff and Grauert . For an alternate proof using resolution of singularities see …
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5
votes
Accepted
Plurisubharmonic function and complete Kähler metric on certain Kähler manifold
Question 1: Plurisubharmonic functions extend across codimension 2 subvarieties . Let X be the complex projective plane blown up at one point and D be the exceptional divisor then any plurisubharmonic …
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5
votes
Is hyperbolicity a Zariski open condition?
Yes,Hyperbolicity is an open condition .You can find it in Brody's paper in Trans Amer Math Soc vol235 1978 page 216 . A more general statement can be found in Kobayashi's book Hyperbolic Complex Spac …
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1
vote
Reference request: Existence of plurisubharmonic potential for positive (1,1)-current on Ste...
The fact that you are looking for requires the current to be exact.On any Stein manifold for which the second cohomology group with integer coefficients has non trivial torsion free part,there are hol …
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11
votes
Accepted
Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?
On page 73 of Kobayashi's book Hyperbolic Complex spaces he shows that if D is a certain configuration of 6 lines in the plane then its complement is complete hyperbolic and hyperbolically embedded i …
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3
votes
Accepted
Connectivity of complements of Stein opens
If you look at corollary 4.10 page 45 of the book of Banica and Stanasila titled Algebraic methods in the global theory of complex spaces,you will find a proof of the following .Any irreducible Stein …
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2
votes
Singularities of the Remmert reduction of a holomorphic convex manifold
The Cartan Remmert reduction can be any normal Stein space.By a theorem of Bingener and Flenner Arch Math (Basel)32(1979) pages 34-37 One can construct normal Stein spaces with unbounded embedding dim …
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