MATH 430/505  Complex Geometry
Spring 2015
Ali Sinan Sertöz
Faculty of Science, Department of Mathematics, Room: SA121
Office Hours: Thursday
13:4015:30
Text Books:
Complex Geometry, An Introduction, Daniel
Huybrechts, Universitext, SpringerVerlag, 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.
Upshot: 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  13:4015:30  SAZ19  
THU  15:4016:30  SAZ19 
Exams and Grading:
Midterm 1  25%  TakeHome  Due on 16 April 2015 Class time  Solution 
Midterm 2  25%  TakeHome  Due on 12 may 2015, Class time  Solution 
Final  35%  TakeHome  20 May 2015 Wednesday, 15:30  Solution 
Homework  15%  TakeHome  

Note: After each takehome work there may be in class quizzes to check your understanding of what you wrote in the takehome work. The questions will be tailored separately for each student. Your questions will be related to those answers you gave in the takehome work. Your grade for that takehome will be determined after this quiz.  
By Yönetmelik Madde 4.7 here is
our FZ grade policy: 
Homework1  Due on 10 February 2015 class time  Solution 
Homework2  Due on 14 April 2015 Class time  Solution 
Homework3  Due on 21 April 2015 Class time  Solution 
Homework4  Due on 13 May 2015 Wednesday  Solution 
Homework5  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
Contact address is: