# Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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### Schemification (schematization?) of locally ringed spaces

Motivation: Say $F: D \to Sch$ is a diagram in the category of schemes, and we're interested in whether it has a colimit (gluings, pushouts, and "categorical" quotients are all examples of colimits)....
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### Mumford-Tate conjecture for mixed Tate motives

Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-...
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### Finiteness of etale cohomology for arithmetic schemes

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$. Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...
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### Homotopy type of the affine Grassmannian and of the Beilinson-Drinfeld Grassmannian

The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$X\mapsto |X^{an}|$$ ...
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### Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians

Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
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### Infinite extensions such that every elliptic curve has finite rank

The comments to this answer seem to make the following claim. Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
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### Does the "holomorphic spheres-to-continuous spheres" forgetful function respect the mixed Hodge structures on homotopy groups?

For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was ...
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### History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
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### The Grothendieck Ring of Higher Stacks

The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that \$[X] = [U] + [Y]...