Talk:Deformation (mathematics)
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Todo
editDiscuss Deformation Functors
editThis page should include a discussion of (pre-)deformation functors with examples. A good resource for this material is https://www.math.ucdavis.edu/~osserman/classes/256A/notes/deform.pdf . Another reference is https://cel.archives-ouvertes.fr/cel-00392119/document
Examples Needed
editIn addition, this should include examples both from geometry and arithmetic:
- Hilbert schemes and deformations
- curves (over finite fields)
- Galois representations
For curves try the following
- take a smooth curve and see if it is smooth over , then take the system with non-changing coefficients
- take a smooth curve over and let one of the coefficients be an infinite series. Each truncation gives a deformation of the curve over .
Differential Graded Lie Algebra
editDiscuss differential graded lie algebras and their use in deformation theory. Look at page 17 of https://arxiv.org/pdf/1409.5996.pdf
Add in Kodaira-Spencer Theory
editThis page should include the basic definitions from Kodaira-Spencer theory, including the
- definition of deformations in the complex-analytic settings
- kodaira-spencer morphism
- vanishing of implies there exists a complete family of deformations where the kodaira-spencer map is an isomorphism
- Add Bogomolev-Tian-Todorov theorem that the Kodaira-Spencer map of a local universal deformation of a Calabi-Yau manifold is an isomorphism. (https://static1.squarespace.com/static/57bf2a6de3df281593b7f57d/t/57bf67bf6a49636398ee220e/1472161728174/tian-todorov.pdf)
Discuss Applications to Singularity Theory
edit- take examples of miniversal deformations from the milnor number page
Formalize KS-Theory for Local Deformations
edit- discuss pro-representability/universal deformation
- give example
- discuss schlessinger's theorem
Hilbert Schemes
edit- discuss how deformation theory is the local study of Hilbert schemes — Preceding unsigned comment added by 97.122.75.155 (talk) 04:47, 2 August 2017 (UTC)
Moduli of Curves/Gromov-Witten Theory
editThis page should also discuss some of the material in http://www.claymath.org/library/monographs/cmim01c.pdf — Preceding unsigned comment added by 161.98.8.4 (talk) 21:19, 30 July 2017 (UTC)
Main theorems in deformation theory
editI think this page should include a survey of some of the main theorems in deformation theory. This should include
- Schlessinger's theorem (give non pro-representable)
- Grothendieck's theorem on deformations of abelian varieties
- Serre-Tate theorem (already there but should be developed further and moved to the theorems section)
- Deligne Illusie theorem
- T^1 lifting theorem
- BTT theorem