MATH 430 / MATH 505
INTRODUCTION TO COMPLEX GEOMETRY
Fall 2010-2011

Ali Sinan Sertöz
Faculty of Science, Department of Mathematics, Room: SA121
Office Hours: Tuesday 10:40-12:30

Synopsis:
The aim of the course is to develop the fundamental tools of complex algebraic geometry that are needed to understand the delicacies of the Hodge Conjecture which is one of the one million Dollar prize problems of the millennium posed by the Clay Mathematics Institute.

The general theme in Hodge conjecture is that certain subsets of a geometric object are also geometric if and only if some algebraic objects associated to them satisfy certain obvious conditions. This was first stated and proved for dimension two case by Lefschetz in 1921. In 1950 Hodge conjectured that this result should be true for any dimension. The conjecture underwent some retuning in the early sixties and remained open ever since.

In the beginning of the twentieth century Hilbert posed twenty three problems which practically shaped the world of mathematical research during that century. With an aspiration to match the fame and fate of that list, in the beginning of the twenty first century the Clay Mathematics Institute posed seven problems as candidates of being guiding engines of research during the new century. Hodge Conjecture is one of these problems. To make matters more appealing, the Clay institute offers a million dollar prize for the resolution of each of these problems. (Let me mention immediately that Perelman who solved the Poincare Conjecture, also one of the million dollar problems, declined to accept the prize saying that everybody knows that his proof is correct and that it is enough for him.)

Syllabus: The following is a tentative plan leading to Hodge Conjecture.

Affine and projective complex spaces, analytic versus polynomial functions and their zero sets, algebra-geometry dictionary.

Holomorphic functions, Hartog's theorem, Weierstrass preparation theorem, GAGA principle.

Complex manifolds, affine and projective.

Some sheaf theory and sheaf cohomology. Cech cohomology.

Vector bundles, connections and characteristic classes.

Kahler geometry.

Dolbaut cohomology, de Rham theorem and Hodge decomposition.

Intersection of cycles on complex manifolds. Chow rings.

Lefschetz's (1,1)-theorem.

Hodge Conjecture and amendments.

Current status of the conjecture.

Prerequisites:

Traditionally, in algebraic geometry you learn what you need as you go along. i.e. Göç yolda toparlanır.
We will follow this tradition and therefore I will not impose any prerequisites on the course.

However a certain degree of curiosity and eagerness to learn and an altruistic quest for knowledge, though not officially required, will be extremely useful in extracting a joy of discovery from the course material.

Both mathematics and physics students, undergraduate or graduate are encouraged to enroll.

Grading will be based on homework and take-home exam performances.

Schedule:

TUE   08:40-10:30 SAZ-19
THU  10:40-12:30 SAZ-19

Sources:

The course will rely on class notes. However sections, and even paragraphs, from the following sources may be useful.
(You can borrow copies of these material from my office anytime.)

• Griffiths and Harris, Principles of Algebraic Geometry, (QA564.G64)
Chapter 0 will be useful.

• Lewis, A Survey of the Hodge Conjecture, Second Edition
The first few chapters may be used. (The library has the first edition which is unreadable.)

• Huybrechts, Complex Geometry, QA641.H89
The first four chapters cover what we want in much more detail than we will.

Though not easily readable for a twenty first century student, here is Hodge's original article where he, albeit indirectly, states the conjecture.

• Hodge, The topological invariants of algebraic varieties,
International Congress of Mathematicians, Cambridge 1950.