All Questions

I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem: Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...

Let $\pi : W \rightarrow Y$ be a holomorphic fibration of complex manifolds. Let $L\rightarrow W$ be a holomorphic line bundle on its total space and denote by $$E^k_q := R^q \pi_*L^k$$ the direct ...

Let $X$ be a complex manifold and let $V\subset X$ be a subvariety. Let $F\rightarrow V$ be a holomorphic vector bundle over $V$ and let $\mathcal{S}=\Gamma(F)$ be the sheaf of holomorphic section of $...

The following question is closely related to this one. Let $X$ a non singular projective variety over $\mathbb C$, and let $\mathscr E$ a locally free sheaf of rank $r$ (an algebraic vector bundle), ...

Let $X$ be a complex manifold and $D$ a normal crossing divisor. Let $U = X - D$ and $j: U \rightarrow X$ the natural map. Voisen observes that there is a natural inclusion $$\Omega^k_X(\log D) \...

Let $X$ a complex manifold, and $\mathcal O_X$ the sheaf of holomorphic functions. Oka Coherence Theorem states that $\mathcal O_X$ is coherent (as $\mathcal O_X$-module). How to prove that also the ...