Math 602D: Topics in Geometry (Fall 2022)

Class Time: MWF 10-11am. AUDX-142

Instructor: Sébastien Picard
Email: spicard@math
Office: MATH 236
Office Hours: MWF 11am-12pm.
Syllabus: Can be found here.

Course Description: This is a course on topics in complex geometry. The first part of the course will be an introduction to complex manifolds and holomorphic vector bundles. The second part of the course will study how objects vary along families of complex manifolds. The third part of the course will focus on Calabi-Yau threefolds, and describe the process of a conifold transition.

References on complex geometry:
- P. Candelas, Lectures on complex manifolds, Superstrings and grand unification. 1988.
- P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley and Sons, 2014.
- D. Huybrechts, Complex geometry: an introduction, Berlin Springer, 2005.
- K. Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der mathmatischen Wissenschaften 1986.
- K. Kodaira and J. Morrow, Complex manifolds, Holt, Rinehart and Winston Inc, 1971.
- G. Szekelyhidi, An introduction to extremal Kahler metrics, AMS Graduate Studies in Mathematics 152 (2014).

-------------
Schedule: (Lecture Notes)

(I) Complex manifolds and holomorphic vector bundles
Sep 7,9: Complex manifolds
Sep 12,14,16: Vector bundles
Sep 21,23: Chern connection
Sep 26,28: Hermitian curvature tensor
Oct 3,5,7: Hodge decomposition
Oct 12,14: Kodaira vanishing theorem
Oct 17,19,21: Sheaves, Lefschetz hyperplane theorem

(II) Deformations of complex manifolds
Oct 24,26,28: Variation of complex structure
Oct 31, Nov 2: Semi-continuity
Nov 7, Nov 14: Kodaira-Spencer stability of the Kahler property

(III) Calabi-Yau threefolds
Nov 16,18: Parameters of threefolds, Ricci-flat metrics
Nov 21,23,25: The quintic example, conifold singularities
Nov 28,30, Dec 2: Candelas-de la Ossa metrics
Dec 12,14,16: Topological transitions of CY3
-------------

Homework:
Homework 1 - Due Oct 3
Homework 2 - Due Oct 24
Homework 3 - Due Nov 21