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Results tagged with complex-geometry
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user 25510
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.
3
votes
Closed analytic subvariety of $\mathbb C^n$ not defined by global holomorphic functions
A closed subset $X$ of $C^n$ satisfying your definition can be defined by global analytic functions. This follows from the solution of the "Second Cousin problem" in $C^n$, and is explained in any tex …
- 91.8k
5
votes
intersection of holomorphic curve with hyperplane
EDITED. Suppose first that $f(C)$ is not contained in any hyperplane.
Let us describe hyperplanes by equations
$L(w)=a_1w_1+\ldots+a_nw_n-b=0$, where we can normalize so that $\sum |a_j|^2+b^2=1$.
Now …
- 91.8k
1
vote
A problem related to connectivity of analytic functions
No, it can be infinite. Start with the function $w=f(z)$ mapping the unit disc onto itself.
Let $I_k$ be disjoint closed arcs on the image disc in the $w$ plane. Deform the image region by adding two …
- 91.8k
1
vote
Riemann surface disconnected at infinity
Let $C$ be a complex line in $C^2$, say $y=0$. Project it on $x$-line, all
properties are satisfied:-)
If you really want "connected, but ONLY if one goes near the origin",
take the set $\{(x,y): y^2 …
- 91.8k
5
votes
Accepted
Is there an explicit formula for the modulus of an annulus given a parameterization of the i...
I think that there is no formula. The best one can do is to estimate. Here is a simpler problem of the same sort: suppose you have a parametrization of the boundary of a simply connected region, and
s …
- 91.8k
5
votes
Accepted
Extension of pluriharmonic functions
Your condition $\dim M>2$ does not save the situation: you can have many counterexamples
with $M=M'\times C^n$ where $\dim M'=1$ and your functions are independent of the second
variable. And in dimen …
- 91.8k
2
votes
Accepted
Bounding the derivative of a holomorphic function on a disk by its absolute value
Yes, this is true. The simple reason is that bounded functions form a normal family. Therefore their derivatives are uniformly bounded on every compact.
To obtain the estimate $|f'(z)|<1$, apply Cauc …
- 91.8k
1
vote
Is there an underlying geometric reason for the rigidity of complex geometry?
To expand my comment, let me mention the wonderful book by R. Penrose, The road to reality. A complete guide to the laws of universe, A.A. Knopf, NY 2006. He discusses this under the title "Magic of c …
- 91.8k
1
vote
Proof verification for a theorem about a harmonic function on the unit disc
This is a consequence of the Phragmen-Lindelof Principle, whose general formulation is the following: Let $D$ be a bounded region
in the plane, and $\zeta_0\in\partial D$. Let $u$ be a subharmonic fun …
- 91.8k
6
votes
An inf-sup estimate for holomorphic functions
This is not true: take $n=1$, $r=1$, $\eta(z)=e^{az},\; a>0,$
then
$$\max_{z\in B(0,r)}|\eta(z)|=e^a,$$
while
$$\min_{z\in B(0,1)}|\eta(z)|=e^{-a}.$$
Since $a>0$ is arbitrary, no $\kappa$ with require …
- 91.8k
15
votes
Accepted
Interesting results for open Riemann surfaces
The results on open Riemann surfaces are not "rare". They are just well forgotten.
I only list a few books which deal with open Riemann surfaces:
MR0114911 (Zbl 0196.33801), MR0228671 (Zbl 0152.27401) …
- 91.8k
1
vote
Higher dimensional analogue of Ahlfors covering surface theory
Yes, there exists such a generalization. It was done by Marie-Helene Schwartz.
Formules apparentées à la formule de Gauss-Bonnet pour certaines applications d’une variété à n dimensions dans une autr …
- 91.8k
3
votes
Examples of pluripolar sets
As you said, analytic sets (other than the whole space) are pluripolar. By playing with subharmnic functions
of the form $\sum u_j$ you can arrange some more complicated sets which are neither open n …
- 91.8k
3
votes
Accepted
Harmonic function osculating a given subharmonic function
The answer is no. The reason is that a subharmonic function does not have to be
continuous. For example, there is a subharmonic function $u$ in the unit disk,
$u(0)=1$ and $u(z_k)=0$ for a sequence $z …
- 91.8k
3
votes
Which are the recommended books for an introductory study of complex manifolds?
MR2359489 Wells, Raymond O., Jr. Differential analysis on complex manifolds. With a new appendix by Oscar Garcia-Prada. Graduate Texts in Mathematics, 65. Springer, New York, 2008.