4452  Differential Geometry
Tuesday 3:00pm  3:50pm, Lunt 103
Thursday 1:00pm  2:40pm, Lunt 103
Course website:
www.math.northwestern.edu/~tosatti/cy.html
Instructor:
Valentino Tosatti
Email: tosatti@math.northwestern.edu
Office: Lunt 225
Office hours:
Monday 3:00pm4:00pm
Course Contents:
This course is an introduction to CalabiYau manifolds.
Topics that will likely be covered include:
 Introduction to Kähler manifolds
 Yau's solution of the Calabi Conjecture
 Applications of the CalabiYau theorem, including:
 Structure theorem for Ricciflat Kähler manifolds
 Chern number inequalities
 Uniqueness of the complex projective plane
 The BogomolovTianTodorov theorem on unobstructed deformations
 K3 surfaces
Prerequisites:
Familiarity with basic differential and Riemannian geometry and complex analysis. We will use some
results about PDE from the course
4201.
Textbook:
We will not follow any textbook directly, but the following references might be useful when studying:
 Z. Błocki, The CalabiYau theorem, Lecture Notes in Mathematics 2038 (2012). [PDF]
 M. Gross, D. Huybrechts, D. Joyce, CalabiYau Manifolds and Related Geometries, Springer, 2003.
 D. Joyce, Compact manifolds with special holonomy, Oxford University Press, 2000.
 Y.T. Siu, Lectures on HermitianEinstein metrics for stable bundles and KählerEinstein metrics, Birkhäuser Verlag, 1987.
[PDF]
The following are some other textbooks that contain basic material on complex and Kähler
manifolds, but which have a possibly different focus:
 W. Ballman, Lectures on Kähler manifolds [PDF]
 S.S. Chern, Complex manifolds without potential theory
 J.P. Demailly, Complex analytic and differential geometry
[PDF]
 A. Futaki, KählerEinstein metrics and integral invariants
[book]
 P. Griffiths, J. Harris, Principles of algebraic geometry
 D. Huybrechts, Complex geometry: an introduction
[book]
 S. Kobayashi, K. Nomizu, Foundations of differential geometry, vol.2
 K. Kodaira, J. Morrow, Complex manifolds
 A. Moroianu, Lectures on Kähler geometry
[PDF]
 G. Tian Canonical metrics in Kähler geoemtry
 R. Wells Differential analysis on complex manifolds
 F. Zheng Complex differential geometry
Grading:
There will be regular homework.
Daily Schedule:
This is a tentative syllabus and it is likely to change as the course progresses.
Date 
Topics Covered 
Remarks 
Jan. 8, 10 
Complex Manifolds 

Jan. 15, 17 
Hermitian metrics 
Homework 1
Due January 22 in class
Solution 
Jan. 22, 24 
Kähler metrics 
Homework 2
Due January 29 in class
Solution 
Jan. 29, 31 
Ricci curvature 
Homework 3
Due February 5 in class
Solution 
Feb. 5, 7 
Proof of the Calabi Conjecture 
Homework 4
Due February 12 in class
Solution 
Feb. 12, 14 
Proof of the Calabi Conjecture 
Homework 5
Due February 19 in class
Solution 
Feb. 19, 21 
Proof of the Calabi Conjecture 
Homework 6
Due February 28 in class
Solution 
Feb. 26, 28 
Chern number inequalities, uniqueness of CP^{2} 
Homework 7
Due March 5 in class
Solution 
Mar. 5, 7 
Albanese map for compact Kähler manifolds 
Homework 8
Due March 12 in class
Solution 
Mar. 12, 14 
Structure theorem for Ricciflat Kähler manifolds 
