# 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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### When is a coherent sheaf on an algebraizable space algebraizable?

Let $\mathcal{X}$ denote an algebraizable rigid analytic space over $\text{Spm}(\mathbb{Q}_p)$,
i.e. there exists a finite type $\mathbb{Q}_p$-scheme $X$ such that $\mathcal{X}$ is its rigid ...

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### GAGA, positive line bundles, Kodaira embedding, and homogeneous coordinate rings

Let $M$ be a compact K"ahler manifold and let $L$ be a positive line bundle over $M$. We know from the Kodaira embedding theorem that from $L^{\otimes k}$ for some $k$ we can construct an ...

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### Is there a translation to English of Serre's GAGA?

I am trying to find an English translation of J.-P. Serre, Géométrie algébrique et géométrie analytique (GAGA). However, I am unable to find a translation searching for it in google. Does anybody know ...

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### Comparing algebraic and analytic spaces through the universal property of classifying topoi

$\newcommand\kAlg{k\mathrm{Alg}}\DeclareMathOperator\Zar{Zar}\newcommand\Mnf{\mathrm{Mnf}}$I apologize beforehand if my question is naïve. I must admit that I do not know much about analytic/smooth ...

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### The notion of border for (complex and non-archimedean) analytic spaces and schemes

Is a manifold with corner an analytic space (just show that $\left[0, +\infty \right)^{n}$ is an analytic space, which seems obvious but maybe I'm wrong...) EDIT: as noted in the comments some complex ...

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### Decomposability and analytification of coherent sheaves

Let $X$ be an affine (algebraic) complex variety and $f:Y \to X$ be a finite morphism. Given any coherent sheaf $\mathcal{F}$ on $X$, we denote by $\mathcal{F}^{an}$ the analytification of $\mathcal{F}...

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### GAGA for vector bundles over Riemann surfaces

Serre’s GAGA theorem gives an equivalence of categories between algebraic and analytic coherent sheaves over a complex projective variety. The proof relies on the finiteness of the cohomologies of ...

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### Kaehler analogue of very ample line bundle

In the correspondence between projective and Kaehler geometry an ample line bundle corresponds to a positive line bundle, where the latter requires that the curvature of the Chern connection is a ...

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### Does the structure morphism matter in GAGA?

Let $X$ be an integral Noetherian separated scheme. Can there exist two morphisms of finite type $X\rightarrow \mathrm{Spec}\,\mathbb{C}$ such that the corresponding complex-analytic spaces are not ...

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### GAGA for stacks

I am curious about stacky generalizations of the following GAGA theorem:
If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is ...

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### 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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### 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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### $\mathscr{D}$-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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### 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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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 $\...

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### 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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### 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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### 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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### 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 ...