Courses WS 17/18

Advanced Geometry I V4D2
Tu 14.15h16h Kleiner HS, Wegelerstrasse 10
Th 12.15h14h SR 0.008
Tutorial: Mo 12.15h14h SR 0.011, Tutor: Johannes Schäfer
Begin: Tuesday, October 10
Topic: Introduction to Kähler geometry
Prerequisits: Geometry I and some familiarity with functions of one
complex variable
A very solid knowledge of calculus in several variables, including
differential forms
Some basic familiarity with de Rham cohomology
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.
Week 1: Basic properties of holomorphic maps; complex
manifolds.
Exercises 1 pdf
The first part of the exercises will be discussed on
Mo, October 16 (and will
not be collected)
The second part of the excercises is due on Oct. 17.
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
Exercises 2 pdf
due on Oct. 24
Please hand in in groups of two or three!
Attention: There will be a lecture on Monday, Oct. 23, at 12.15pm
The tutorial is moved to Thursday, Oct. 26
Week 3: Chern connection and its curvature, examples,
line bundles
p.3132, p.1415, Section 3.4 in Ballmann's book
Exercises 3 pdf
due on Nov. 2
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
Exercises 4 pdf
due on Nov.7
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
Exercises 5 pdf
due on Nov. 14
Week 6: Various characterization of Kähler manifolds, Examples
Chapter 3 in Voisin's book
Part of Chapter 4 in Ballmann's book (Hopf manifold)
Exercises 6 pdf
due on Nov. 21
Attention: Problem 3 will be postponed to the following week
Week 7: Positivity of line bundles revisited.
Kähler metrics on blowups of points and on projective
bundles
Chapter 3 in Voisin's book
Exercises 7 pdf
due on Nov. 28
Attention: Problem 2 will be postponed to the following week
Week 8: The class will be taught by Bram Petri.
The excercise sheet 8 will be available on his homepage.
Week 9: There will be a class on Monday, Dec. 4, at 12.15h
The class on Thursday, Dec. 7, will be replaced by
a tutorial
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)
Exercises 9 pdf
due on Dec. 12
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.
Exercises 10 pdf
due on Dec. 19
Week 11: The Riemann Roch theorem
The Kodaira
embedding theorem for surfaces.
Chapter 7 of Donaldson's book, chapter 9 of Jost's book
Exercises 11 pdf
these exercises will not be collected
Happy holidays!
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
Exercises 12 pdf
due on Jan. 16
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
Exercises 13 pdf
will not be collected
Final examen: Tuesday, Jan. 30, at 2.00pm,
Kl. Hörsaal
Office hours: Friday, Jan. 19, 2pm3pm,
Wednesday, Jan. 24, 2pm3pm
You may bring one sheet of paper (DIN A4) with formulas etc.
Courses SS 17

Advanced Geometry II V4D4
Tu 14h16h SR 1.007
We 12h14h SR 1.007
Tutorial: Mo 10h12h SR 1.007, Tutor: Gabriele Viaggi
Begin: Tuesday, April 18
Topic: The ending lamination theorem for hyperbolic 3manifolds
of positive injectivity radius
Prerequisits: Solid knowledge in basic differential geometry and basic
topology
Literature:
There are many books which cover basic properties of hyperbolic
manifolds
Examples of such books are
R. Benedetti, C. Petronio, Lectures on hyperbolic geometry,
Springer Universitext, Springer 1992.
P. Buser, Geometry and Spectra of compact Riemann surfaces,
Birkhäuser Progress in Math. 1992.
B. Iversen, Hyperbolic geometry, London Math. Society Student text 25,
1992.
S. Katok, Fuchsian groups, Chicago Lectures in Math., Chicago 1992.
The book project
B. Martelli, An introduction to geometric topology,
arXiv:1610.02592
contains a large amount of material related to the class.
B. Maskit, Kleinian groups, Springer Grundlehren der Math. 287,
Springer 1988.
K. Matsuzaki, M. Taniguchi, Hyperbolic manifolds and
Kleinian groups, Oxford Sci. Publ, Oxford 1998.
M. Bridson, Haefliger, Metric spaces of nonpositive
curvature, Springer Grundlehren 319, Springer 1999.
Week 1: The geometry of the hyperbolic plane (the first chapter
in Katok's book)
Exercises 1 pdf
These exercises will be discussed on Mo, April 24 (and will
not be collected)
Week 2: Closed hyperbolic surfaces, simple closed
geodesics,
the GaussBonnet formula
Chapter 1 of Katok's book and Chapter 2.4 and 3 of Buser's book are
good sources
Exercises 2 pdf
Due on May 3
Attention: This sheet does not coincide with the sheet
that was posted on April 23
Week 3: Right angled hyperbolic hexagons, hyperbolic
pairs of pants, pair of pants decompositions of
closed hyperbolic surfaces.
The collar theorem and
the theorem of Bers.
The material can be found in Section 4.1, Section 3.1
(Prop. 3.1.8) and Section 5.2 of Buser's book
Exercises 3 pdf
Due on May 10
Week 4: Area and short closed geodesics on a hyperbolic
surface. Hyperbolic 3space
The material can be found in Section 5.2 of Buser's book
and in Chapter 2 of the book draft of Martelli.
Exercises 4 pdf
Due on May 17
Week 5: Hyperbolic 3space and its isometry group.
The hyperboloid model and the action of the isometry
group on $S^2$.
The material can be found
in Chapter 2 of the book draft of Martelli
Exercises 5 pdf
Due on May 24
Week 6: Constructing hyperbolic 3manifolds by bending.
Quasigeodesics and geodesics,
precisely invariant sets and proper actions.
Chapter VIII.E. of Maskit's book discusses the
examples, but the viewpoint is different
Exercises 6 pdf
Due on May 31
Week 7: Constructing quasigeodesics from piecewise
geodesics; more on proper actions.
Limit set
Most can be found in Maskit's book.
Exercises 7 pdf
Due on June 14
Week 8: Domains of discontinuity and limit sets
in Section 4.1.1 of the book draft
Geometry on groups; basic properties of hyperbolic spaces
The book by Bridson and Haefliger (see p.140) contains
all relevant facts.
Exercises 8 pdf
Due on June 21
Week 9: Proper hyperbolic spaces and their boundaries
The material is explained ion p.427432 in the book by
Bridson and Haefliger
Characterization of quasifuchsian groups
Section 4.1.2 in the book draft
Exercises 9 pdf
Due on June 28
Attention: There will be a class
on Mo, June 26, at 10.15h and on Tuesday, June 27, at 14.15h
The class on June 28 and the tutorial is cancelled
Exercises 10 pdf
Due on July 5
Week 12: Ends of 3manifolds
Closed geodesics exiting an end
Length and intersection of closed geodesics
The material can be found in the
book draft
Exercises 11 pdf
Due on July 19
Attention: There will be a class
on Mo, July 17, at 10.15h.
The class on Tuesday, July 18, at 14.15h is replaced
by the tutorial
Office hour: Mo, July 24, 14h15h
Notes (will be expanded with the class)
pdf
Book draft (version July 10, will be expanded with the class)
pdf
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