Courses SS 18

Advanced Geometry II V4D4
Mo 10.15h12h, SR 0.008
We 10.15h12h SR 0.006
Tutorial: Tu 12.15h14h, SR 1.007, Tutor: Andreas Pieper
Begin: April 16, 2018
Topic: Kähler geometry
Prerequisits:
Most of the material of the class on Advanced Geometry I
as summarized below
Exercise sheet 1
This sheet will be discussed on April 24 and will not be collected.
Week one: Sheaves
Exercise sheet 2
Due on April 25
Exercise sheet 3
Due on May 2

Graduate seminar on differential geometry S4D2
with Werner Ballmann and Anna Siffert
More information
Organizational meeting: April 9, 2018
Courses WS 17/18

Advanced Geometry I V4D2
Second final: Tuesday, March 6,
oral examination
Topic: Introduction to Kähler geometry
Summary:
Week 1: Basic properties of holomorphic maps; complex
manifolds.
Week 2: Holomorphic vector bundles,
the holomorphic tangent bundle,
types of differential forms, Chern connection
Sections 2.2 and 3.2 in Ballmann's book
Week 3: Chern connection and its curvature, examples,
line bundles
p.3132, p.1415, Section 3.4 in Ballmann's book
Week 4: Divisors and line bundles, the Picard group, curvature
Section 3.4 in Ballmann`s book, Section 5.6 in Jost's book
Section 2.2 in Huybrechts's book contains a purely sheaftheoretic
account which I did not pursue
Week 5: Degree of divisors and line bundles on Riemann surfaces
Relation to curvature
Kähler manifolds: Characterization by LeviCivita and Chern connection
Section 5.6 in Jost's book
Proposition 2.18 (p.20) in Ballmann's book (the signs in the proof are wrong)
p.4647 in Ballmann's book (the formula on p.47 is wrong)
Chapter 3 in Voisin's book is recommended for reading
Week 6: Various characterization of Kähler manifolds, Examples
Chapter 3 in Voisin's book
Part of Chapter 4 in Ballmann's book (Hopf manifold)
Week 7: Positivity of line bundles revisited.
Kähler metrics on blowups of points and on projective
bundles
Chapter 3 in Voisin's book
Week 8: Hodge star operator and beyond
Week 9:
The Hodge star operator, Dolbeault cohomology, the
Laplacian, harmonic (p,q), forms, formulation of the
main theorem of Hodge theory
Everything can be found in Griffith Harris's book (which is full of
small mistakes)
Week 10: Applications of the Hodge theorem to compact Riemann surfaces:
Computing the dimension of the vector space of
holomorpic oneforms, construction of meromorphic functions.
A good source is Chapter 7 of Donaldson's book.
Week 11: The Riemann Roch theorem
The Kodaira
embedding theorem for surfaces.
Chapter 7 of Donaldson's book, chapter 9 of Jost's book
Week 12: Cohomology of Kähler manifold
Prop. 3.14, Section 6.1.1, 6.1.2 and part of 6.1.3 of Voisin's book
Week 13: Hodge theory for differential form with values
in a holomorphic line bundle
The Kodaira vanishing theorem
Section 3 and 5 of Ballmann's book
Week 14: The Kodaira embedding theorem
Found in the book by Griffith and Harris
Literature:
W. Ballmann, Lectures on Kähler manifolds, Eur. Math. Soc. 2006
D. Huybrechts, Complex geometry, Springer
R. Narasimhan, Several complex variables, Chicago Lect. in Math. 1971.
R.O.~Wells, Differential Analysis on Complex Manifolds,
Springer Graduate Texts in Math. 65 (1980)
Chapter IX of S. Kobayashi, K. Nomizu,
Foundations of Differential Geometry II, 1969.
J. Jost, Compact Riemann Surfaces, Universitext, Springer
P. Griffith, J. Harris, Principles of Algebraic Geometry,
Wiley 1978.
C. Voisin, Hodge Theory and Complex Algebraic Geometry I,
Cambridge Univ. Press 2002.
R. Bott, L. Tu, Differential forms in algebraic topology,
Springer Grad. Text, 1982.
S. Donaldson, Riemann surfaces, Oxford Grad. Text in Math. 22, 2011.
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