An introduction to C-infinity schemes and C-infinity algebraic geometry

Journal article

Abstract:: If X is a smooth manifold then the R-algebra C^∞(X) of smooth functions c : X → R is a "C∞-ring". That is, for each smooth function ƒ : Rⁿ → R there is an n-fold operation Φƒ : C^∞(X)ⁿ → C^∞(X) acting by Φƒ: (c₁,...,c_n) |→ f(c₁,...,c_n), and these operations Φƒ satisfy many natural identities. Thus, C^∞(X) actually has a far richer structure than the obvious R-algebra structure.

We explain a version of algebraic geometry in which rings or algebras are replaced by C^∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are "C^∞-schemes", a category of geometric objects generalizing manifolds, and whose morphisms generalize smooth maps. We also discuss "C^∞-stacks", including Deligne-Mumford C^∞-stacks, a 2-category of geometric objects generalizing orbifolds. We study quasicoherent and coherent sheaves on C^∞-schemes and C-infinity stacks, and orbifold strata of Deligne-Mumford C^∞-stacks. This enables us to use the tools of algebraic geometry in differential geometry, and to describe singular spaces such as moduli spaces occurring in differential geometric problems.

Copyright holder:: International Press
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