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 …

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 …

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 …

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 …

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 …

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- …

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 …

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 …

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 …

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 …

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 …

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] …

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 …

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)= …

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 …