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I am learning about Hilbert scheme of points $S^{[n]}$ on projective K3 surfaces S. Since these are hyperkähler varieties, the second cohomology $H^2(S^{[n]},\mathbb{Z})$ is endowed with the non-degen …

I have the following basic question. Everything is over $\mathbb{C}$. Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow …

Given a holomorphic vector bundle $E$ on a compact complex Kähler manifold $X$ (I am happy to assume $X$ projective), we can compute the sheaf cohomology $H^\ast(E)$ of $E$ using the Dolbeault complex …

Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection. It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e …

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$. Question: What can we say about the algebra structure of …