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 -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