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Questions tagged [gaga]

GAGA is short for Serre's 1956 paper "Géometrie Algébrique et Géométrie Analytique". The tag refers not only to that paper, but also to the way of thinking introduced by it.

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415 views

Defining algebraic manifold without referring to schemes

Let $M$ be a complex manifold admitting an atlas with each chart biholomorphic to $\mathbb{C}^n$ and transition maps being rational functions. Is it true that there exists a smooth integral ...
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0answers
131 views

Algebrizing analytic quotients of algebraic spaces

Suppose that $Y$ is an algebraic space over $\mathbb{C}$ which is "nice" (say, separated and of finite type). Let $R\subset Y\times Y$ be a closed algebraic subspace which forms a categorical ...
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0answers
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$\mathscr{D}$-module external tensor product and analytification

Let $X$ and $Y$ be smooth complex varieties and $M\in D^b_{\mathrm{hol}}(\mathscr{D}_X)$ resp. $N\in D^b_{\mathrm{hol}}(\mathscr{D}_Y)$ holonomic $\mathscr{D}$-modules (resp. complexes with holonomic ...
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0answers
178 views

Comparison of algebraic and analytic q-expansion

I would like to check that algebraic and analytic q-expansion of a modular form coincide. I'm thinking about modular forms as global sections of some sheaf on modular curves. If $X$ is a modular ...
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0answers
213 views

Does the cohomology comparison part of GAGA hold over the reals?

If $k=\mathbb{C}$, $X$ is a projective (or even proper) scheme over $k$ and $F$ a coherent sheaf on $X$, then there is an isomorphism $$ H^p(X,F)\rightarrow H^p(X^{an}, F^{an}) $$ (and there is also ...
2
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0answers
132 views

Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable ...
11
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3answers
1k 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 cohomology $H^*_{...
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0answers
206 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 each algebraic ...
7
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1answer
953 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 torus is algebraic ...
6
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1answer
747 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 space), and let $\...
22
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1answer
3k 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 polynomials and ...
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4answers
4k 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 agrees with the ...
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2answers
1k 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 equivalent to an affine ...
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1answer
2k 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 implies affine, but ...