Math 430/505
Introduction to Complex Geometry
Spring 2020
Ali Sinan Sertöz
Office: Faculty of Science, Department of Mathematics, Room: SA121
Phone: 290 1490
Office Hours: Thursday 10:40-12:30
Schedule:
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)
Attendance
Higher Education Council (YÖK) requires us to take regular attendance. Therefore I will take attendance but this will not affect your grade.
Textbooks:
Griffiths and Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978 (download book pdf)
Huybrechts, Complex Geometry, Springer-Verlag, 2005, (download book pdf)
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.
Grading:
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.
Office: Faculty of Science, Department of Mathematics, Room: SA121
Phone: 290 1490
Office Hours: Thursday 10:40-12:30
Schedule:
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)
Attendance
Higher Education Council (YÖK) requires us to take regular attendance. Therefore I will take attendance but this will not affect your grade.
Textbooks:
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)Prerequisites
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.
Grading:
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.
| [0,40) | F |
| [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 |
| 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 |
2-12 |
| 10 Feb, 13 Feb |
Affine and projective spaces, varieties,
implicit function theorem, complex manifolds, real and
complexified tangent spaces |
12-22 |
| 17 Feb, 20 Feb |
De Rham and Dolbeault cohomology,
calculus on complex manifolds, Stokes' theorem |
23-34 |
| 24 Feb, 27 Feb |
Sheaves and Cech cohomology, the de Rham
and Dolbeault theorems, Leray theorem, some calculations
of cohomology |
34-49 |
| 2 Mar, 5 Mar |
Intersection theory of cycles, Poincare
duality |
49-65 |
| 9 Mar, 12 Mar |
Vector bundles, connections and
curvature, hermitian metric |
66-80 |
| 19 Mar |
Harmonic theory on compact complex
manifolds |
80-100 |
| 23 Mar, 26 Mar |
Applications of the Hodge theorem |
100-106 |
| 30 Mar, 2 Apr |
Kahler manifolds, Hodge identities, Hodge decomposition | 106-118 |
| 6 Apr, 9 Apr |
Lefschetz decomposition, hard Lefschetz
theorem |
118-126 |
| 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 |
148-156 |
| 4 May, 7 May |
Lefschetz theorem on hyperplane sections |
156-161 |
| 11 May, 14 May |
Lefschetz (1,1)-theorem and the Hodge
conjecture |
161-164 |
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.
