Formal geometry and deformation theory, winter 2017-18

Organization

Time Monday, 10:15-11:45
October 16 until February 12
Location FU Berlin, A3/SR210
Organizers Tanya Kaushal Srivastava
Pedro A. Castillejo

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.

Schedule

October 16 Tanya (+ a little bit Pedro) Introduction Notes
October 23 Tanya Formal geometry I: Locally Noetherian Formal Schemes
October 30 Tanya Formal Geometry II: The Comparison Theorem
November 6 Yun Formal Geometry III: Grothendieck Existence Theorem
November 13 Julian Deformation Theory I: Infinitesimal Deformations I
November 20 Marcin Deformation Theory II: Infinitesimal Deformations II
November 27 Marco Deformation Theory III: Infinitesimal Deformations III
December 4 Efstathia Deformation Theory IV: Formal Deformations I
December 11 tba Deformation Theory V: Formal deformations II
December 18 Marcin Deformation Theory VI: Formal vs Algebraic Deformation.
January 8 Xiaoyu Su Lifting theory I: Lifting from Char p to Char 0
January 15 Yun Lifting theory II: Serre's Example of non-liftable variety
January 22 Fei Lifting theory III: Lifting Abelian varities from Char p to Char 0
January 29 Holger Elbe Special session of rigid geometry and Berkovich spaces
February 5 Simon and Pedro From formal to rigid geometry I (Admissible blow ups)
February 12 Simon and Pedro From formal to rigid geometry II (Raynaud's fiber functor)

Literature

Here is the list of references.