Talk:Deformation (mathematics)

Todo

Discuss Deformation Functors

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

Examples Needed

In 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 $\mathbb {Z} /q^{2}$, 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 $\mathbb {F} _{p}$.

Differential Graded Lie Algebra

Discuss 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

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 Applications to Singularity Theory

• take examples of miniversal deformations from the milnor number page

Formalize KS-Theory for Local Deformations

• discuss pro-representability/universal deformation
• give example
• discuss schlessinger's theorem

Hilbert Schemes

• 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

This 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

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

Some references

• http://web.archive.org/web/20151010220650/https://math.berkeley.edu/~achinger/notes/char-p-5.pdf
• https://math.stackexchange.com/questions/102252/english-translation-or-summary-of-relevements-modulo-p2-et-decomposition-du

Kaptain-k-theory (talk) 04:19, 20 August 2021 (UTC)