## Tagged Questions

1answer
159 views

### algebraic de Rham cohomology of singular varieties

Hi, Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular …
0answers
126 views

### How do fibers of the functor Algebraic Varieties $\to$ Complex Analytic Spaces look like?

There's already a question (which got several interesting answers) asking about examples of the phenomenon of non (essential) injectivity of the functor $U:Alg\to AnEsp$, mapping e …
1answer
260 views

### Are complex varieties Kahler? - Algebraic, non-projective complex manifolds

Let $X/\mathbb{C}$ a nonsingular proper variety and $X_{an}$ it's associated analytic space. Is $X_{an}$ necessarily Kahler? Certainly we know this if $X$ is projective. A complex …
1answer
475 views

### What is the intuition behind the proof of the algebraic version of Cartan’s theorem A?

I am trying to understand the idea behind the proof of GAGA. A crucial step is the following: Theorem: Let $X=\mathbb{P}^r_{\mathbb{C}}$ (either as a variety or as an analytic spa …
4answers
2k views

### Algebraic de Rham cohomology vs. analytic de Rham cohomology

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 …
1answer
1k views

### GAGA and Chern classes

My question is as follows. Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant pol …
2answers
757 views

### Topologically contractible algebraic varieties

From a post to The Jouanolou trick: Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally eq …
1answer
1k views

### Stein Manifolds and Affine Varieties

When is a Stein manifold a complex affine variety? I had thought that there was a theorem saying that a variety which is Stein and has finitely generated ring of regular functions …
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