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
Deformation Theory, Hartshorne
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
Monday 12/11 - The cotangent complex's climactic conclusion