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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.
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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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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14
votes
Accepted
A geometric characterization of smooth points of a complex algebraic variety
The answer to all three of your questions is yes.See the book by E M Chirka titled
Complex Analytic Sets pages 189,190 and 120 .These questions are local so this is true on Kahler manifolds .
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13
votes
Accepted
Kähler manifold which is not algebraic
generic complex tori in complex dimension 2 or higher.
MR
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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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9
votes
$\partial \bar{\partial}$ lemma for contractible domains
If in addition you assume that your domain is pseudoconvex then by a theorem of
A.Aeppli what you want is true.The paper is titled :On the cohomology structure of Stein
manifolds 1965 (Proc.Conf.Comp …
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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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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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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
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
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Your assumption implies that there is a finite holomorphic map of the universal cover of X into complex euclidean space .In particular the universal cover is Stein.
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5
votes
Accepted
A corollary to Stone-Weierstrass theorem
In your case we can find a holomorphic function on the plane that uniformly approximates the
given continuous function .It is a consequence of the following .Suppose K is a compact measure zero subset …
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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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4
votes
Accepted
The Levi form of the distance squared function in a non-positively curved Kaehler manifold
By the Hessian comparison theorem the square of the distance function on X is strictly convex.
On Kahler manifolds strictly convex functions are strictly plurisubharmonic .By Grauert's
solution of th …
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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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