Math 430/505
Introduction to Complex Geometry

Spring 2020

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Ali Sinan Sertöz
Office: Faculty of Science, Department of Mathematics, Room: SA121
Phone: 290 1490
Office Hours: Thursday 10:40-12:30

MON 10:40-12:30 SA-Z02
THU  08:40-10:30
SA-Z02 (later in the semester we may cancel the 08:40 class)

Higher Education Council (YÖK) requires us to take regular attendance. Therefore I will take attendance but this will not affect your grade.

Griffiths and Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978  (download book pdf)
Huybrechts, Complex Geometry, Springer-Verlag, 2005, (download book pdf)
We will cover most of the first 164 pages of Griffiths and Harris' book. We will use parts of Huybrechts' book. (see the syllabus below)
Traditionally, in algebraic geometry you learn what you need as you go along. i.e. Göç yolda toparlanır.
I will start with the basics of single variable complex analysis, pass to several variables and before we know it we will be doing complex geometry!
Though no technical prerequisite is emposed, a sincere desire to learn for the sake of learning will be most helpful. :)
Take the study of geometry in the vein of  Ars Gratia Artis.

Course Contents:
Our concrete aim in this course is to understand the content, meaning and the significance of the Hodge Conjecture which is one of the Millenium Prize Problems.
In geometry we inevitably assign some algebraic entities to geometric spaces so that we can do calculations. Sometimes there are more algebraic structures than that make sense geometrically. For example when you take the square root of the square of 1, you end up with an extra -1 which you did not start with. Hodge conjecture states that such an anomaly does not occur if we work on a complex projective manifold. What makes the conjecture mystic is that it does not hold on any nice manifold except those which are projective. We will learn all about this story in this course.

The grading will be through take-home exams administered several times during the semester. Sincere attempts to understand the solutions will be more valuable than exact solutions borrowed from other sources. I include the following grading catalogue for completeness; grading will be generous provided that you hand in each take-home exam on time. At the end of the year depending on class performance the following catalogue may be relaxed.

[40,45) D
[45,50) D+
[50,53) C-
[53,55) C
[55,57) C+
[57,60) B-
[60,65) B
[65,68) B+
[68,71) A-
[71,100] A

Past Complex Geometry Courses:
Fall 2010    Spring 2015    Spring 2017

Exam 1
Due: 12 March 2020
Exam 2
Due: 20 April 2020
Exam 3
Due: 8 may 2020
Exam 4
Due: 12 June 2020

SYLLABUS for Math 430/505 Spring 2020
3 Feb, 6 Feb
Rudiments of complex analysis, single and several complex variable theory, Cauchy formula, Hartogs' theorem
10 Feb, 13 Feb
Affine and projective spaces, varieties, implicit function theorem, complex manifolds, real and complexified tangent spaces
17 Feb, 20 Feb
De Rham and Dolbeault cohomology, calculus on complex manifolds, Stokes' theorem
24 Feb, 27 Feb
Sheaves and Cech cohomology, the de Rham and Dolbeault theorems, Leray theorem, some calculations of cohomology
2 Mar, 5 Mar
Intersection theory of cycles, Poincare duality
9 Mar, 12 Mar
Vector bundles, connections and curvature, hermitian metric
19 Mar
Harmonic theory on compact complex manifolds
23 Mar, 26 Mar
Applications of the Hodge theorem
30 Mar, 2 Apr
Kahler manifolds, Hodge identities, Hodge decomposition 106-118
6 Apr, 9 Apr
Lefschetz decomposition, hard Lefschetz theorem
13 Apr, 16 Apr
Divisors and line bundles, Chern classes of line bundles 129-146
20 Apr
Adjunction formulas 146-148
27 Apr, 30 Apr
Kodaira vanishing theorem
4 May, 7 May
Lefschetz theorem on hyperplane sections
11 May, 14 May
Lefschetz (1,1)-theorem and the Hodge conjecture

The third column shows the corresponding page numbers of Griffiths and Harris' book on which the lecture material will roughly be based.
We will not follow the book verbatim; we will skip some too technical parts and spend more time on understanding the crucial ideas.
You can consult Huybrechts' book from time to time to see a different way of explaining the same ideas. Griffiths and Harris' book has no exercises. We will invent our own exercises and from time to time borrow some from Huybrechts' book.