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Ciao Francesco! The answer to your question is yes, and the work of Bogomolov you are looking for about semistability of the tangent space (for minimal surfaces of general type indeed, no need of ampl …

I assume that for $\operatorname{Hom}(X,Y)$ you mean $\operatorname{Hol}(X,Y)$, that is the family of all holomorphic maps from $X$ to $Y$, endowed with its universal complex structure (which exists s …

Let me write this too long comment as an answer. As abx says, what we do know is Theorem 1. If a Kähler manifold $X$ is homeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to it. This is due to …

At least in the projective setting the following holds true (this is taken from J. Kollár "Shafarevich maps and automorphic forms", Proposition 13.14.2). Proposition. Let $X$ be a smooth projective va …

This question is somewhat related to this other question of mine. I was wondering which are the known examples of bounded domains $\Omega$ in $\mathbb C^n$ admitting a compact free quotient. By a theo …

Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is c …

If $X$ is smooth, and if you ask to have the same homotopy type of a CW complex of real dimension at most $n$, this is precisely the statement of the Andreotti-Frankel theorem. It is true, more gener …

Maybe this can be of some help. In this paper by K. Oguiso, you can find at the end an appendix by N. Nakayama. In this appendix he gives a theorem which describes the local structure of a degenerati …

Just for fun, here is an answer in the purely algebraic setting. So, suppose that $X$ is irreducible projective algebraic of dimension $n$, $\alpha=c_1(\mathcal O_X(D))$ is the class of a nef diviso …

Of course it depends mostly on your background. But the first chapter, as well as the first half of the second chapter of Kollár-Mori's "Birational Geometry of Algebraic Varieties" is an incredibly in …

Kobayashi hyperbolicity (or Brody hyperbolicity, the two notions coincide for compact complex spaces), is an open condition with respect to the analytic topology. Thus, for instance, once you find an …

I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3. Warning (in view of the comment below by S.S.): the holonom …

This follows by the following general, quite elementary Fact. Suppose you have a complex vector bundle $E\to X$ on a compact manifold $X$ and two hermitian form $h_1$ and $h_2$ on $E$, such that th …

Here is how it works for the (complex) Grassmannian. I will leave you the pleasure to extend this point of view to others homogeneous spaces (for instance complete and incomplete flag manifolds). Fir …

A large class of compact complex manifolds for which (more generally) $$ \operatorname{Aut}(X)=\operatorname{Bim}(X) $$ holds is given by Kobayashi hyperbolic compact complex spaces. Here $\operatorn …