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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.
List of contents
1. Introduction; 2. Background on C¿-schemes 3. Background on manifolds with (g-)corners; 4. (Pre) C¿-rings with corners; 5. C¿-schemes with corners; 6. Boundaries, corners, and the corner functor; 7. Modules, and sheaves of modules; 8. Further generalizations and applications; References; Glossary of Notation; Index.
About the author
Kelli Francis-Staite read for her DPhil at the University of Oxford as a Rhodes Scholar. Her thesis developed the theory of C∞-algebraic geometry with corners. She is currently an Adjunct Senior Lecturer at the University of Adelaide.Dominic Joyce is Professor of Mathematics at Oxford University and a Senior Research Fellow at Lincoln College Oxford. He is the author of 'Compact Manifolds with Special Holonomy' (2000), 'Riemannian Holonomy Groups and Calibrated Geometry' (2007), 'A Theory of Generalized Donaldson-Thomas Invariants' (2012 co-authored with Yinan Song), and 'Algebraic Geometry over C∞-rings' (2019). Joyce is winner of the LMS Whitehead and Fröhlich prizes, an EMS prize, the Adams prize, and he is a Fellow of the Royal Society.
Summary
Crossing the boundary between differential and algebraic geometry, the authors introduce algebro-geometric methods into differential geometry, allowing differential geometers to study singular or infinite-dimensional spaces. In particular the authors discuss 'C8-schemes with corners', differential-geometric spaces with a good notion of boundary.
