## Publication Date:

1 July 2019

## Journal:

Memoirs of the American Mathematical Society

## Last Updated:

2021-06-24T14:14:11.037+01:00

## Volume:

260

## DOI:

10.1090/memo/1256

## abstract:

If $X$ is a smooth manifold then the $\mathbb R$-algebra $C^\infty(X)$ of

smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each

smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation

$\Phi_f:C^\infty(X)^n\to C^\infty(X)$ acting by $\Phi_f:(c_1,\ldots,c_n)\mapsto

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

$\mathbb R$-algebra structure.

We develop a version of algebraic geometry in which rings or algebras are

replaced by $C^\infty$-rings. As schemes are the basic objects in algebraic

geometry, the new basic objects are $C^\infty$-schemes, a category of

geometric objects which generalize smooth manifolds, and whose morphisms

generalize smooth maps. We also study quasicoherent and coherent sheaves on

$C^\infty$-schemes, and $C^\infty$-stacks, in particular Deligne-Mumford

$C^\infty$-stacks, a 2-category of geometric objects generalizing orbifolds.

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. This paper forms the foundations of the

author's new theory of "derived differential geometry", surveyed in

arXiv:1206.4207 and in more detail in arXiv:1208.4948, which studies

d-manifolds and d-orbifolds, "derived" versions of smooth manifolds and

smooth orbifolds. Derived differential geometry has applications to areas of

symplectic geometry involving moduli spaces of $J$-holomorphic curves.

Many of these ideas are not new: $C^\infty$-rings and $C^\infty$-schemes have

long been part of synthetic differential geometry. But we develop them in new

directions. This paper is surveyed in arXiv:1104.4951.

smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each

smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation

$\Phi_f:C^\infty(X)^n\to C^\infty(X)$ acting by $\Phi_f:(c_1,\ldots,c_n)\mapsto

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

$\mathbb R$-algebra structure.

We develop a version of algebraic geometry in which rings or algebras are

replaced by $C^\infty$-rings. As schemes are the basic objects in algebraic

geometry, the new basic objects are $C^\infty$-schemes, a category of

geometric objects which generalize smooth manifolds, and whose morphisms

generalize smooth maps. We also study quasicoherent and coherent sheaves on

$C^\infty$-schemes, and $C^\infty$-stacks, in particular Deligne-Mumford

$C^\infty$-stacks, a 2-category of geometric objects generalizing orbifolds.

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. This paper forms the foundations of the

author's new theory of "derived differential geometry", surveyed in

arXiv:1206.4207 and in more detail in arXiv:1208.4948, which studies

d-manifolds and d-orbifolds, "derived" versions of smooth manifolds and

smooth orbifolds. Derived differential geometry has applications to areas of

symplectic geometry involving moduli spaces of $J$-holomorphic curves.

Many of these ideas are not new: $C^\infty$-rings and $C^\infty$-schemes have

long been part of synthetic differential geometry. But we develop them in new

directions. This paper is surveyed in arXiv:1104.4951.

## Symplectic id:

195876

## Download URL:

## Submitted to ORA:

Submitted

## Publication Type:

Journal Article