Talk:Deformation (mathematics)

Todo

This 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

In addition, this should include examples both from geometry and arithmetic:

For curves try the following

take a smooth curve and see if it is smooth over $\mathbb {Z} /q^{2}$ , then take the system with non-changing coefficients
take a smooth curve over $\mathbb {Z} _{p}$ and let one of the coefficients be an infinite series. Each truncation gives a deformation of the curve over $\mathbb {F} _{p}$ .

Discuss differential graded lie algebras and their use in deformation theory. Look at page 17 of https://arxiv.org/pdf/1409.5996.pdf

This 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 $H^{2}(X,T_{X})$ 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 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)[reply]

I 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