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



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.


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 -Smoothness of \(\text{Hilb}^n(S)\) for \(S\) a surface, continued; intro to Tian-Todorov

Monday 10/2 - Deformations of smooth varieties; pro-representability. Applications to curves and deformations of affine varieties.

Wednesday 10/4 - T1-lifting theorem, beginning of proof of Tian-Todorov

Monday 10/9 - Tian-Todorov, continued.  Intro to generic vanishing.

Wednesday 10/11 - The derivative complex; the tangent cone theorem.

Monday 10/16 - Proof of the tangent cone theorem; Hodge theory for line bundles

Wednesday 10/18 - Proof of generic vanishing theorem, following Green-Lazarsfeld

Monday 10/23 - Unobstructedness for Abelian varieties

Wednesday 10/25 - NO CLASS

Monday 10/30 - More on Abelian varieties, deforming with polarization, algebraization

Wednesday 11/1 - Formal GAGA

Monday 11/6 - NO CLASS, university holiday

Wednesday 11/8 - Finish formal GAGA, applications

Monday 11/13 - Bend and break, intro

Wednesday 11/15 - More bend and break

Monday 11/20 - Isomonodromic deformations

Wednesday 11/22 - NO CLASS, Thanksgiving Break

Monday 11/27 - NO CLASS

Wednesday 11/29 - NO CLASS

Monday 12/4 - Deforming singular schemes; the naive cotangent complex

Wednesday 12/6 - The cotangent complex, in all its glory

Monday 12/11 - The cotangent complex's climactic conclusion