Math 666, Fall 2007:
Complex differential geometry
Meeting MWF 11:30-12:20 in Milner 216

Instructor: Joseph (JM) Landsberg

Office: Milner 329

Phone: (979)-458- 0625
E-mail: jml@math.tamu.edu
office hours: Tues 1-2pm, Wed. 2-3 pm or by appointment.

Syllabus and text:  We will cover  chapters 1-6 in
Voisin, Claire Hodge theory and complex algebraic geometry. I. Translated from the French original by Leila Schneps. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2002. x+322 pp. ISBN: 0-521-80260-1
(also available much cheaper in French: Théorie de Hodge et géométrie algébrique complexe.  Cours Spécialisés [Specialized Courses], 10. Société Mathématique de France, Paris, 2002. viii+595 pp. ISBN: 2-85629-129-5 )
supplemented by lecture notes of Demailly (long version) (click here for more concise Demailly notes) and Siu on complex manifolds and analysis.
In addition there will be four topics: i. a detailled proof of the Kodaira embedding and vanishing theorems, ii.
an overview  of Douglas' work on complex geometry and operator theory, iii.

Prerequisite: a first course in graduate differential geometry (definitions of manifolds, vector bundles)

 
Homework:  There will occasional homework problems of varying
level of difficulty. All but the most basic problems will be optional.

Exams: There will be no exams.

Grading policy: adaptable to the needs of the students.

Tentative syllabus ("chapter x" refers to chapter x of Voisin):

week 1: overview of the course, results from complex analysis  (chapter 1)
week 2: definitions and first properties of complex manifolds (chapter 2)
week 3: Kahler geometry: basic definitions, first properties,
connections. (chapter 3)
week 4: sheaves (chapter 4)
week 5: sheaf cohomology (chapter 4, cont'd)
week 6: Laplacians and their uses (chapter 5)
week 7: Kahler manifolds (chapter 6)
week 8: Kodaira embedding and vanishing theorem (from Demailly notes and Griffiths and Harris)
week 9: Kodaira theorems and consequences
week 10: complex geometry and operator theory (work of Ron Douglas)
week 11: complex geometry and operator theory (cont'd)
week 12: deformations of complex manifolds and Kahler manifolds
week 13: hyperbolicity (following these Demailly notes)

click here for a precise description of what is covered in each lecture

 
Homework assignments:
These are basic problems - I will also provide challenging problems
with no fixed due date on a regular basis.
due 9/5: problems 1-3 p37 of Voisin
due 9/12: problems 1,2 p61 of Voisin
due 9/17: problems 1,2 p 82 of Voisin
due 9/28: this set of three problems
due 10/22: this set of three problems
due 10/29: exercises from chapter 4 of Voisin.
due 11/26: LAST ASSIGNMENT!
 exercise (2) from chapter 5 of Voisin,
 exercise (2) from chapter 6 of Voisin,
 all chapter 7 exercises from Voisin,
and give a short direct proof that any two different hermitian metrics
on a holomorphic line bundle L-> X have cohomologous curvature tensors (considering
the curvature as a 2-form on X)
Note: There will be no classes Sept. 19,21,24,26  and Nov. 21
these missed classes will be made up Wed. 4-5:20pm on the
following days: 10/31, 11/7, 11/14 in Milner 216
 

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