Math 524, Transcendental Algebraic Geometry (a.k.a. Linear Analysis on Manifolds)
My information:
Thomas Nevins
357 Altgeld Hall
217.265.6762
nevins@uiuc.edu
Office hours: Wednesdays, 4-5 p.m., and by appointment
Prerequisites: Math 520 (Differentiable Manifolds), or reasonable familiarity with the basics of differentiable manifolds and (singular) cohomology, and Math 542 (Complex Variables) or reasonable familiarity with some basics of
complex analysis.
Text: D. Huybrechts, "Complex geometry: an introduction," Springer (Universitext),
2005.
An electronic version is available
here
(the UIUC library has purchased electronic access).
Other references:
- D. Arapura, "Complex algebraic varieties and their cohomology"
(on-line here).
- B. Berndtsson, "L2-methods for the
∂-equation"
(on-line here).
- J.-P. Demailly, "Complex analytic and differential geometry"
(on-line here).
- P. Griffiths and J. Harris, "Principles of algebraic geometry."
- C. Voisin, "Hodge theory and complex algebraic geometry" (two volumes: vol. 1 will be more relevant).
- R.O. Wells, "Differential analysis on complex manifolds."
General
AG References
This course will give an introduction to some methods of complex geometry, with applications to complex projective manifolds (and,
more generally, Kahler manifolds).
Complex projective manifolds, a.k.a. smooth complex projective varieties, are objects of crucial interest to
algebraic geometers but also provide a storehouse of important, relatively concrete examples for complex differential geometry. Such manifolds come naturally
equipped with special metrics known as Kahler metrics. The existence of a Kahler metric on a compact complex manifold leads to a wealth of new information,
much of it encoded in special ways in the cohomology of the manifold.
We will develop the basics of compact Kahler manifolds and applications to geometric
and topological invariants. We will also develop a number of interesting and important examples: for example, algebraic curves, abelian varieties, complete
intersections.
The course will assume that students have a basic working familiarity with
differentiable manifolds and complex variables, and some acquaintance
with singular cohomology, but we will not assume any very deep knowledge of these
subjects. Contrary to what the title given in the course catalog may suggest, this will not be a course that develops the theory of linear differential operators on manifolds in great detail. We will, however, at the very least sketch proofs of some important theorems (most notably, the Hodge Theorem) by analytic methods that are ubiquitous in complex geometry (for example, via linear elliptic theory or, alternatively, via heat equation techniques).
Lecture 1
Lecture 2
Lecture 3
Lectures 4-5
Lecture 6
Lecture 7
Lectures 8-9
Lectures 10-12
Lectures 13-14
Lectures 15-17
Lectures 18-20
Lectures 21-25
Lecture 26
Lecture 27
Lectures 28-30
Lecture 31
Lectures 32-33
Lecture 34
Hodge numbers, part I
Hodge numbers, part II
Homework 1 [latex]
(due Wednesday, September 10).
Homework 2 [latex]
(due Monday, September 22).
Homework 3 [latex]
(due Friday, October 10).
Homework 4[latex]
(due November 5).
Homework 5[latex]
(to be posted).
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