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

Abstract

This is a survey of the author's paper arXiv:1001.0023 on Algebraic Geometry over If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a C-infinity ring. That is, for each smooth function f : R^n --> R there is an n-fold operation \Phi_f : C^\infty(X)^n --> C^\infty(X) acting by \Phi_f: (c_1,...,c_n) |--> f(c_1,...,c_n), and these operations \Phi_f satisfy many natural identities. Thus, C^\infty(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 rings. As schemes are the basic objects in algebraic the new basic objects are C-infinity schemes, a category of geometric objects generalizing manifolds, and whose morphisms generalize smooth maps. We also discuss C-infinity including Deligne-Mumford stacks, a 2-category of geometric objects generalizing orbifolds. We study quasicoherent and coherent sheaves on schemes and stacks, and orbifold strata of Deligne-Mumford stacks. This enables us to use the tools of algebraic geometry in differential and to describe singular spaces such as moduli spaces occurring in differential geometric problems. Many of these ideas are not new: rings and schemes have long been part of synthetic differential geometry. But we develop them in new directions. In a new book, surveyed in arXiv:1206.4207 and at greater length in arXiv:1208.4948, the author uses algebraic geometry to develop a theory of differential geometry, which studies d-manifolds and d-orbifolds, derived versions of smooth manifolds and orbifolds. D-orbifolds will have applications in symplectic as the geometric structure on moduli spaces of J-holomorphic curves.