0
votes
1answer
237 views
Dolbeault cohomology
Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}} (X)$ and
$H^{(0,1)}_{\bar{\partial}}(X)$ . I don't know how to do this but if we use Kunnet formula we …
0
votes
0answers
46 views
On uniform convergence of sequences of bounded holomorphic functions with formal convergence
At some point I needed to prove that some formal (iterative) construction yielded an actual convergent power series. To do so I was led to prove the following lemma, where $\mathca …
0
votes
0answers
45 views
Conformal properties of complex Schwarz-Christoffel tranformations
Let $a_1< a_2<\dots < a_n$ be $n$ real numbers and assume that
$\beta_1,\dots,\beta_n\in \mathbb R$ are such that $\sum_i \beta_i=n-2$.
In this case, it is well-knonw th …
0
votes
1answer
197 views
Stein manifolds definiton
There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions …
1
vote
3answers
175 views
polynomial zero within a square
Is there any complex polynomial $p$ of one variable having no zeros within the unit square:
$-1 < \Re(z) , \Im(z) < 1$ such that
$\left|p(0)\right|$ is strictly smaller than …
0
votes
0answers
34 views
Conformal properties of complex Schwarz-Christoffel maps
Let $a_1< a_2 < \cdots < a_n$ be $n$ real numbers
assume that $\beta_1,\ldots,\beta_n\in \mathbb R$ are such that $\sum_i \beta_i=n-2$.
In this case, it is well- …
0
votes
1answer
206 views
Question on Hartogs’s Extension Theorem
Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)?
For Hartogs's Extension Theorem see here:
http://en …
1
vote
0answers
61 views
Analytical continuation of electrostatic potentials
I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the …
6
votes
2answers
386 views
Reason for studying coherent sheaves on complex manifolds.
Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf $F$ on a complex manifold $X$ i …
3
votes
2answers
214 views
j-invariant duplication, triplication and quintuplication formulae… how?
I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the correspondin …
1
vote
0answers
82 views
Question about a oscillatory integrals on manifold
Let $M$ be a compact oriented Riemannian manifold without boundary.
Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$,
where $a(x),b(x)$ are real-valued function o …
0
votes
0answers
116 views
What are zeros of certain entire functions?
How to describe the zeros of an entire function $\sin(\pi z) +\int_0^1 \exp (izt) d\mu(t)$,
where a complex-valued measure $\mu$ satisfies $\mu ({0 }) = \mu ({1 }) =0, |\mu|([0,1] …
2
votes
2answers
101 views
Weierstrass factorization with $L^2$ estimates?
Let $\Omega$ be a bounded domain in $\mathbb{C}$. Let $X$ be a discrete set of points whose boundary is in the boundary of $\Omega$. Can I find an $L^2$ holomorphic function whic …
1
vote
0answers
65 views
Univalent functions with non-negative coefficients
Is anything non-trivial known about univalent functions with non-negative coefficients?
Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)= …
1
vote
0answers
110 views
The relation between the weak Lefschetz theorem and the strong Lefschetz theorem
The Weak Lefschetz Theorem states that for a compact Kahler manifold, $Pic(X) \rightarrow H^{1,1}(X, \mathbb{Z})$ is surjective.
The Hard Lefschetz Theorem states that for a comp …

