MATH 430/505 - Complex Geometry
Spring 2015

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

Office Hours:
Thursday 13:40-15:30

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:
 TUE 13:40-15:30 SAZ-19 THU 15:40-16:30 SAZ-19

 Midterm 1 25% Take-Home Due on 16 April 2015 Class time Solution Midterm 2 25% Take-Home Due on 12 may 2015, Class time Solution Final 35% Take-Home 20 May 2015 Wednesday, 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 10 February 2015 class time Solution Homework-2 Due on 14 April 2015 Class time Solution Homework-3 Due on 21 April 2015 Class time Solution Homework-4 Due on 13 May 2015 Wednesday Solution Homework-5 Due on 14 May 2015 Thursday Solution

The course will be graded according to  the following catalogue:

 [0,39] F [40,44] D [45,49] D+ [50,53] C- [54,56] C [57,59] C+ [60,62] B- [63,65] B [66,68] B+ [69,71] A- [72,100] 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 3, 5 Feb Several complex variables 2 10, 12 Feb Complex manifolds 3 17, 19 Feb Affine varieties 4 24, 26 Feb Projective varieties 5 3, 5 Mar Sheaf theory 6 10, 12 Mar Cech cohomology 7 19 Mar de Rham cohomology 8 24, 26 Mar Dolbeault cohomology 9 31 Mar, 2 Apr Algebraic cycles 10 7, 9 Apr Poincare duality 11 14, 16 Apr Harmonic forms 12 21 Apr Hodge decomposition 13 28, 30 Apr Hodge conjecture - popular version 14 5, 7 May Hodge conjecture - general version 15 12, 14 May What is known about the Hodge conjecture

Old Exams are on Old Courses Web Page