4451  Differential Geometry
Monday, Wednesday, Friday 1:00pm  1:50pm, Lunt 101
Course website:
www.math.northwestern.edu/~tosatti/cdg.html
Instructor:
Valentino Tosatti
Email:
tosatti@math.northwestern.edu
Office: Lunt 225
Office hours:
Thursdays 1.00pm  2.00pm
or by appointment
Course Description:
This course is an introduction to complex analytic techniques in algebraic geometry.
Topics that will likely be covered include:
 Plurisubharmonic functions and closed positive currents
 Lelong numbers
 Hörmander's L^{2} estimates for ∂
 OhsawaTakegoshi extension theorem
 Demailly's regularization theorem for currents
 Multiplier ideal sheaves and Nadel vanishing theorem
 Applications to algebraic geometry:
 Invariance of plurigenera
 Fujita's approximation theorem
 BoucksomDemaillyPăunPeternell's characterization of uniruled manifolds
 Openness conjecture
Prerequisites:
Familiarity with basic differential geometry and complex analysis.
Textbook:
We will roughly follow the textbook:
 J.P. Demailly, Analytic Methods in Algebraic Geometry, International Press, 2012
[PDF]
Two other useful references are
 J.P. Demailly, Complex analytic and differential geometry
[PDF]
 R. Lazarsfeld, Positivity in algebraic geometry, I and II, Springer, 2004.
Daily Schedule:
This is a tentative syllabus and it is likely to change as the course progresses.
Date 
Topics Covered 
Remarks 
Sept. 21, 23, 25 
Plurisubharmonic functions 

Sept. 28, 30, Oct. 2 
Closed positive currents, Lelong numbers 

Oct. 5, 7, 9 
Hörmander's L^{2} estimates for ∂ 

Oct. 12, 14, 16 
FriedrichsHörmander's density in the graph norm 

Oct. 19, 21, 23 
OhsawaTakegoshi extension theorem 

Oct. 26, 28, 30 
Demailly's regularization theorem for currents 

Nov. 2, 4, 6 
Multiplier ideal sheaves and Nadel vanishing theorem 

Nov. 9, 11, 13 
Invariance of plurigenera 

Nov. 16, 18, 20 
Openness conjecture, Volume of line bundles 

Nov. 23, 25 
Fujita's approximation theorem 

Nov. 30, Dec. 2 
BoucksomDemaillyPăunPeternell's characterization of uniruled manifolds 
