MATH 430/505 - Complex Geometry
Spring 2017

Ali Sinan Sertöz
Faculty of Science, Department of Mathematics, Room: SA-121

Office Hours:
Wednesday 10:40-11:30

Text Books:
Complex Geometry, An Introduction, Daniel Huybrechts, Universitext, Springer-Verlag, 2005. (first three chapters)

Principles of Algebraic Geometry, Philip Griffiths and Joseph Harris, Wiley Interscience, 1978. (parts of chapter 0)
A Survey of the Hodge Conjecture, James D. Lewis, CRM Monograph Series Vol: 10, American Mathematical Society, 1991. (selected sections)
The Hodge Theory of Projective Manifolds, Mark Andrea de Cataldo, Imperial College Press, 2007. (to be read only after the course is over)

Synopsis:
Using complex numbers in geometry opens up unexpected and surprising new avenues of exploration. We will weave the course around the famous Hodge conjecture so that at any moment we will have a solid motivation to embrace the new techniques and concepts that arise.

For an extended synopsis you may want to read this summary.

For a brief history of the Hodge conjecture you may read this.

And finally, here is a synopsis of Hodge theory by Emre Sertöz who took this course back in 2010:

• Modulo details, integrating a differential form $\omega$ over a submanifold $N$ of $M$ corresponds to intersecting $N$ with a submanifold $L_\omega$ of $M$, depending only on $\omega$, and counting the number of points of intersection! This is Poincare  duality from differential topology.

•  There is a certain numerical criterion satisfied by $L_\omega$ if it is a complex submanifold of a complex manifold $M$.

•  Hodge conjecture tells us that this numerical criterion is sufficient! Any differential form having this numerical criterion corresponds to a complex submanifold.

• Intersection theory of complex submanifolds is better behaved than the real case. Think of intersecting a line with a curve. Number of intersection points is reduced to counting the number of roots of a polynomial over $\mathbb{C}$ (immediate!) as opposed to $\mathbb{R}$ (good luck!).

• The polynomial analogy hits the mark. Complex submanifolds often end up being cut out by polynomials (in  the meromorphic functions of the ambient space). Intersecting complex submanifolds reduces to solving these polynomials simultaneously.

• Up-shot: We essentially reduce integration of a differential form over a submanifold to counting the number of roots of a system of polynomials in many variables! Hodge theory tells us precisely when we can do this.

• Back to reality: Performing the integration might be easier than calculating the number of solutions of the polynomials involved!! So not all integrals are reduced to finger counting. But being able to shift your perspective so drastically is to be coveted.

Also the web page of the course in 2010 contains some more information.

Schedule:
 WED 08:40-10:30 SAZ-02 FRI 11:40-12:30 SAZ-02

 Midterm 1 25% Take-Home Due on 17 March 2017 Class time Solution Midterm 2 25% Take-Home Due on 28 April 2017, Class time Solution Final 35% Take-Home 26 May 2017 Friday, 15:30 Solution Homework 15% Take-Home Note: After each take-home work there may be in class quizzes to check your understanding of what you wrote in the take-home work. The questions will be tailored separately for each student. Your questions will be related to those answers you gave in the take-home work. Your grade for that take-home will be determined after this quiz. By Yönetmelik Madde 4.7  here is our FZ grade policy: Failure to attend at least 50% of the classes or averaging less than 40% from the two midterms will result in an FZ grade.

 Homework-1 Due on 17 February 2017 Class time Solution Homework-2 Due on 3 March 2017 Class time Solution Homework-3 Due on 31 March 2017 Class time Solution Homework-4 Due on 14 April 2017 Class time Solution Homework-5 Due on 12 May 2017 Class time Solution

The course will be graded according to  the following catalogue:

 [0,39] F (39,44] D (44,49] D+ (49,53] C- (53,56] C (56,59] C+ (59,62] B- (62,65] B (65,68] B+ (68,71] A- (71,100] A

Syllabus:

 The topics below will be discussed only to the extend that they are required to understand the statement and immediate verifications of the Hodge conjecture. Week Date Subjects to be covered 1 8, 10Feb Several complex variables 2 15, 17 Feb Complex manifolds 3 22, 24 Feb Affine varieties 4 1, 3 Mar Projective varieties 5 8, 10 Mar Sheaf theory 6 15, 17 Mar Cech cohomology 7 22, 14 Mar de Rham cohomology 8 29, 31 Mar Dolbeault cohomology 9 5, 7 Apr Algebraic cycles 10 12, 14 Apr Poincare duality 11 19, 21 Apr Harmonic forms 12 26, 28 Apr Hodge decomposition 13 3, 5 May Hodge conjecture - popular version 14 10, 12 May What is known about the Hodge conjecture

Old Exams are on Old Courses Web Page