Talk:Deformation (mathematics)

Latest comment: 3 years ago by Kaptain-k-theory in topic Main theorems in deformation theory

Todo

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Discuss Deformation Functors

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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

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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  , 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

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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

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This page should include the basic definitions from Kodaira-Spencer theory, including the

Discuss Applications to Singularity Theory

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  • take examples of miniversal deformations from the milnor number page

Formalize KS-Theory for Local Deformations

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

Hilbert Schemes

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Moduli of Curves/Gromov-Witten Theory

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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)Reply

Main theorems in deformation theory

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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

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