Fall 2017 - topics in Algebraic geometry

deformation theory

course description

This course is an introduction to deformation theory, at the level of Hartshorne's book on the subject.   Possible topics would include: Schlesinger's criterion, the Tian-Todorov theorem on deformations of Calabi-Yau manifolds, the cotangent complex, infinitsemial variations of Hodge structure, liftability of K3 surfaces, etc.  Familiarity with algebraic geometry at the level of Harshorne's "Algebraic Geometry" is a sufficient prerequisite.

The course philosophy will be to alternate between theory (e.g. Schlesinger's criterion) and applications which use that theory.  We will cover a different application each week or two.

Time and place

11:40am-12:55pm, Room 407 in the Columbia math department, Mondays and Wednesdays.  Some special classes Fridays; see the calendar below.

Recommended texts

FGA Explained 

Deformation Theory, Hartshorne

SGA 2

Notes

Henry Liu is taking notes on this course.  Please let either him or me know if you spot any errors.

Office hours

Office hours by appointment.  I will also hold extra sessions some Fridays; see the calendar below.

Calendar

Wednesday 9/6 - Introduction, tangent-obstruction theories

Monday 9/11 - Examples: deformation of line bundles, application to Lefschetz theorem for Picard groups, deformations of representations

Wednesday 9/13 - Deformation rings, Schlessinger's criterion

Friday 9/15 (room 528, 1-2pm) - Schlessinger's criterion continued, dimensions of deformation rings

Monday 9/18 - NO CLASS

Wednesday 9/20 - NO CLASS

Monday 9/25 - Smoothness of \(\text{Hilb}^n(S)\) for \(S\) a surface

Wednesday 9/27 -The \(T1\)-lifting theorem

Monday 10/2 - The Tian-Todorov Theorem (unobstructedness of Calabi-Yau manifolds)