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Expanding the comment of Donu Arapura, let $X$ be a variety and $Y\subset X$ a subvariety. Then, you have a short exact sequence of sheaves $$ 0\to\mathcal I_Y\to\mathcal O_X\to\mathcal O_X/\mathcal I …

Let $S$ be a smooth projective $K3$ surface, say over the complex numbers, and suppose that $S$ admits a (non-constant) fibration $\pi\colon S\to C$ over a curve $C$. By the universal property of th …

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 …

Let $X$ be a Calabi-Yau threefold. From a complex analytic point of view, it is widely believed that it should not be Kobayashi hyperbolic, that is it should always admit some non-constant entire map …

Let $X$ be a compact complex manifold of complex dimension $n$. The Hodge-Frölicher spectral sequence starts with $$ E_1^{p,q}=H^{p,q}(X,\mathbb C) $$ and the limit term $E^{p,q}_\infty$ is the grade …

As Alexander Woo said in a comment, toric varieties are rational. Now, it turns out that projective hypersurfaces have strong hyperbolicity-type properties. This properties have been established by se …

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 …

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 …

There are lots of references. Mainly every textbook which treats Hodge theory. Try to look at: Voisin: Hodge theory and complex algebraic geometry. I Huybrechts: Complex geometry Wells: Differential …

Concerning your example, there is definitely no analytic proof of the existence of rational curves on Fano manifolds. This is one of the dream of complex geometers... You can also consider this weaker …

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 …

Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify …

The Bloch conjecture states the following: Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim X$ …

The conjecture Francesco is referring to as the "hyperbolicity conjecture" is actually the Green-Griffiths-Lang conjecture. It states that on any given smooth projective manifold of general type $X$ t …

The Fubini-Study metric on $\mathbb P^n$ arises as the curvature of a line bundle. More precisely, once you fix a hermitian inner product in the complex vector space you are projectivizing, you get a …