The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the s …

Suppose $M$ is a complex manifold and $\Omega$ a (edit: bounded) pseudoconvex domain in $M$. Let $u:M\setminus\Omega\to\mathbb{R}$ be a pluriharmonic function. Is it true that $u$ …

The classical Global Residue theorem is formulated as follows (cf. Griffiths & Harris): Let $M$ be a closed complex manifold of dimension n. Then for a meromorphic n-form $\om …

Does this outline of a proof work? Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorph …

I have 3 general abstract reasons to care about complex analysis in a single variable: The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, …

Is there any sequence $ \\{ Z_{\nu} \\}_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in …

I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here i …

Let $Q=[0,\infty)\times [0,\infty)\subset \mathbb C$ and $f: Q\times Q\to Q\times Q$ be a diffeomorphism. such that $f$ is holomorphic in the interior of $Q\times Q$. Can we exte …

A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restriction …

Cartan's theorem B states that every coherent analytic sheaf on a Stein manifold is acyclic. Hartshorne claims that the converse to this theorem also holds, but doesn't supply a r …

I am starting to dive into a study of Several Complex Variables. I would like to have a few guiding examples of "big payoffs". These should be very natural sounding theorems whic …

Let $\Omega$ be a pseudo convex domain. Let $r$ be any $C^2$ function $r: \mathbb C^2\to \mathbb R$. Let $\Omega: \{z: r(z)<0\}$. Then we know that $\psi: -log(-r)+\lambda |z|^ …