Let $X$ be a nice variety over $\mathbb{C}$, where nice probably means smooth and proper. I want to know: How can we show that the hypercohomology of the algebraic de Rham complex …

Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and …

Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest po …

This question is related to a previous question of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein. P …

I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical ca …

I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roug …

Why is it that the vanishing of the integral first Chern class of a compact Kahler manifold is equivalent to the canonical bundle being trivial? I can see that it implies that the …

Let $G$ be a Lie Group and $H$ a closed subgroup such that $G/H$ (the set of right cosets) is a complex manifold manifold. Now $\Omega^1(G/H)$, the space of complex one forms, is a …

In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\w …

Possible Duplicate: Two definitions of Calabi-Yau manifolds Given a compact kahler manifold M with diffeomorphically trivial canonical bundle. Is it true that the canonica …

Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where $R_i=\opl …

This question may be trivial, I did not think hard about it. A friend of mine is looking for an irreducible (reduced) analytic subspace $C \subset \mathbb{C}^2$ with the followin …

I am in this, rather friendly, situation: I have a complex projective space $\mathbb{P}^n$, and there i have a (possibly non-smooth) hypersurface $S$ defined by one irreducible po …

Let $V$ be a complex vector space of dimension 6 and let $G\subset {\mathbb P}^{14}\simeq {\mathbb P}(\Lambda^2V)$ be the image of the Plucker embedding of the Grassmannian $Gr(2, …

As can be guessed from some of my previous questions, I'm trying to understand, at the moment, the relationship between principal and their associated vector bundles. To this end I …