8 - Further generalizations and applications

Summary

We discuss four generalizations and applications of $C^\infty$-schemes with corners.

1. 1. To provide a model for a theory of ‘Synthetic Differential Geometry with corners’.

2. 2. To a theory of ‘ $C^\infty$-stacks with corners’. $C^\infty$-Stacks are studied in D. Joyce, ‘Algebraic Geometry over $C^\infty$-rings’, Memoirs of the AMS, 2019. Most of the theory extends to corners with no changes.

3. 3. To a theory of ‘ $C^\infty$-schemes with a-corners’. ‘Manifolds with a-corners’ are introduced in D. Joyce, arXiv:1605.05913 as a class of manifolds with corners with an alternative smooth structure, that has applications in analysis, e.g. Morse theory moduli spaces should really be manifolds with a-corners, not corners.

4. 4. To a theory of ‘derived $C^\infty$-schemes and derived $C^\infty$-stacks with corners’, and within these, ‘derived manifolds with corners’ and ‘derived orbifolds with corners’, where ‘derived’ is in the sense of Derived Algebraic Geometry.

These have important applications in Floer theories and areas of Symplectic Geometry involving moduli spaces of J-holomorphic curves, as such moduli spaces should be derived orbifolds with corners (or Kuranishi spaces with corners).