Schemes in algebraic geometry can have singular points, whereas differential geometers typically focus on manifolds which are nonsingular. However, there is a class of schemes, 'C∞-schemes', which allow differential geometers to study a huge range of singular spaces, including 'infinitesimals' and infinite-dimensional spaces. These are applied in synthetic differential geometry, and derived differential geometry, the study of 'derived manifolds'. Differential geometers also study manifolds with corners. The cube is a 3-dimensional manifold with corners, with boundary the six square faces. This book introduces 'C∞-schemes with corners', singular spaces in differential geometry with good notions of boundary and corners. They can be used to define 'derived manifolds with corners' and 'derived orbifolds with corners'. These have applications to major areas of symplectic geometry involving moduli spaces of J-holomorphic curves. This work will be a welcome source of information and inspiration for graduate students and researchers working in differential or algebraic geometry.
Author(s): Kelli Francis-Staite
Edition: 1
Publisher: Cambridge University Press
Year: 2024
Language: English
Pages: 222
Introduction
Background on C∞-schemes
Introduction to category theory
C∞-rings
Modules and cotangent modules of C∞-rings
Sheaves
C∞-schemes
Complete C∞-rings
Sheaves of Oₓ-modules on C∞-ringed spaces
Sheaves of Oₓ-modules on C∞-schemes
Applications of C∞-rings and C∞-schemes
Background on manifolds with (g-)corners
Manifolds with corners
Monoids
Manifolds with g-corners
Boundaries, corners, and the corner functor
Tangent bundles and b-tangent bundles
Applications of manifolds with (g-)corners
(Pre) C∞-rings with corners
Categorical pre C∞-rings with corners
Pre C∞-rings with corners
Adjoints and (co)limits for pre C∞-rings with corners
C∞-rings with corners
Generators, relations, and localization
Local C∞-rings with corners
Special classes of C∞-rings with corners
C∞-schemes with corners
(Local) C∞-ringed spaces with corners
Special classes of local C∞-ringed spaces with corners
The spectrum functor
C∞-schemes with corners
Semi-complete C∞-rings with corners
Special classes of C∞-schemes with corners
Fibre products of C∞-schemes with corners
Boundaries, corners, and the corner functor
The corner functor for C∞-ringed spaces with corners
The corner functor for C∞-schemes with corners
The corner functor for firm C∞-schemes with corners
The sheaves of monoids Mᵉˣ,Mⁱⁿ on C(X)
The boundary ∂X and k-corners Cᵏ(X)
Fibre products and corner functors
B- and c-transverse fibre products in Manᵍᶜⁱⁿ, C∞Schᶜᵗᵒ
Modules, and sheaves of modules
Modules over C∞-rings with corners, (b-)cotangent modules
(B-)cotangent modules for manifolds with (g-)corners
Some computations of b-cotangent modules
B-cotangent modules of pushouts
Sheaves of Oₓ-modules, and (b-)cotangent sheaves
(B-)cotangent sheaves and the corner functor
Further generalizations and applications
Synthetic Differential Geometry with corners
C∞-stacks with corners
C∞-rings and C∞-schemes with a-corners
Derived manifolds and orbifolds with corners
Bibliography
Glossary of Notation
Index
