Formal geometry and deformation theory, winter 2017-18
Here is the preliminary program.
The topic of this seminar is formal geometry and deformation theory. The idea of the seminar is to get familiarity with formal schemes by studying its basic properties and using them in order to understand different problems.
The first aim of the seminar is to understand Raynaud's generic fiber functor, which establishes an equivalence between some formal schemes and rigid analytic spaces or Berkovich spaces, depending on the chosen approach.
The second aim of the seminar is to work a little bit with deformation theory. The idea is to study the infinitesimal and formal deformations of a geometric object (for example, two lines deform to an hyperbola), and apply the techniques to some problems lifting from positive characteristic to zero characteristic: for example, we can always lift curves and abelian varieties, in the sense that we find one in characteristic zero such that its special fiber is the one we started with. But, in general, we can't always do this, as we will see with the example done by Serre. All these things will be formulated in the language of formal schemes.
||Tanya (+ a little bit Pedro)
||Formal geometry I: Locally Noetherian Formal Schemes
||Formal Geometry II: The Comparison Theorem
||Formal Geometry III: Grothendieck Existence Theorem
||Deformation Theory I: Infinitesimal Deformations I
||Deformation Theory II: Infinitesimal Deformations II
||Deformation Theory III: Infinitesimal Deformations III
||Deformation Theory IV: Formal Deformations I
||Deformation Theory V: Formal deformations II
||Deformation Theory VI: Formal vs Algebraic Deformation.
||Lifting theory I: Lifting from Char p to Char 0
||Lifting theory II: Serre's Example of non-liftable variety
||Lifting theory III: Lifting Abelian varities from Char p to Char 0
||Special session of rigid geometry and Berkovich spaces
||Simon and Pedro
||From formal to rigid geometry I (Admissible blow ups)
||Simon and Pedro
||From formal to rigid geometry II (Raynaud's fiber functor)
Here is the list of references.