Dokuz Eylül University Information Package / Courses Catalog

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type Of Course	D	U	L	ECTS
MAT 5013	COMPLEX GEOMETRY	ELECTIVE	3	0	0	8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSISTANT PROFESSOR CELAL CEM SARIOĞLU

Offered to

Mathematics
Mathematics

Course Objective

The aim of the course is to develop the fundamental tools of complex analytic geometry, complex algebraic geometry and complex differential geometry.

Learning Outcomes of the Course Unit

1	will be able to explain the properties of holomorphic functions of single and several variables.
2	will be able explain the properties of Kähler manifolds.
3	will be able to understand the geometry of complex manifolds.
4	will be able to know Sheaf Cohomology.
5	will be able to know Hodge theory and current statement of the Hodge conjecture.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week	Subject	Description
1	Holomorphic functions of one variables.
2	Holomorphic functions of several variables.
3	Complex manifolds, affine and projective spaces.
4	Holomorphic vector bundles.
5	Divisors and Line bundles. Blow-up s.
6	Kähler metrics and Kähler manifolds.
7	Sheaves and Cohomology.
8	Sheaves and Cohomology. + Midterm
9	Harmonic forms and Cohomology.
10	Hodge theory on Kähler manifolds.
11	Lefschetz theory.
12	Hermittian vector bundles and Serre duality. Connections, Curvature, and Chern classes.
13	Applications of Cohomology: Hirzebruch-Riemann-Roch theorem, Kodaira vanishing theorem, Kodaira embedding theorem.
14	Deformations of complex structures.

Recomended or Required Reading

Textbooks:
1. Daniel Huybrechts, Complex Geometry an introduction, Springer, 2005.
2. Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge University Press, 2002.
Supplementary Books:
3. Jean-Pierre Demailly, Complex Analytic and Differential Geometry. The lecture notes available at the web site https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
4. Phillip Griffits, Joseph Harris, Principles of Algebraic Geometry, Wiley publishing, 1984.
5. A. Moroianu, Lectures on Kähler geometry, London Mathematical Society Student Texts 69, Cambridge University Press, 2007.
6. Klaus Fritzsche, Hans Grauert, From Holomorphic Functions to Complex Manifolds, Springer, 2002.
7. Donu Arapura, Algebraic Geometry over the Complex numbers, Springer, 2012.
References:
8. Claire Voisin, Hodge Theory and Complex Algebraic Geometry II, Cambridge University Press, 2003.
9. Kunihiko Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer, 1986.
10. James D. Lewis, A survey of the Hodge conjecture, 2nd edition, American Mathematical Society, 1999.

Planned Learning Activities and Teaching Methods

Lecture notes, Presentation, Problem solving, Homework Assignments

Assessment Methods

SORTING NUMBER	SHORT CODE	LONG CODE	FORMULA
1	ASG	ASSIGNMENT
2	MTE	MIDTERM EXAM
3	FIN	FINAL EXAM
4	FCG	FINAL COURSE GRADE	ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5	RST	RESIT
6	FCGR	FINAL COURSE GRADE (RESIT)	ASG * 0.30 + MTE * 0.30 + RST * 0.40

*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be succesful, at the end of the term, the relative grade must be 75 or greater.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

Asst. Prof. Dr. Celal Cem Sarıoğlu
E-mail: celalcem.sarioglu@deu.edu.tr
Office: +90 232 301 8607

Office Hours

TBA

Work Placement(s)

None

Workload Calculation

Activities	Number	Time (hours)	Total Work Load (hours)
Lectures	13	3	39
Preparations before/after weekly lectures	13	4	52
Preparation for midterm exam	1	25	25
Preparation for final exam	1	30	30
Preparing assignments	6	8	48
Midterm	1	3	3
Final	1	3	3
TOTAL WORKLOAD (hours)			200

Contribution of Learning Outcomes to Programme Outcomes

PO/LO	PO.1	PO.2	PO.3	PO.4	PO.6	PO.8	PO.9	PO.10	PO.11
LO.1	3	4	4	4	3	4	3	4	3
LO.2	3	4	4	4	3	4	3	4	3
LO.3	3	4	4	4	3	4	3	4	3
LO.4	3	4	4	4	3	4	3	4	3
LO.5	3	4	4	4	3	4	3	4	3

COURSE UNIT TITLE

DEGREE PROGRAMMES

Description of Individual Course Units

Offered By

Level of Course Unit

Course Coordinator

Offered to

Course Objective

Learning Outcomes of the Course Unit

Mode of Delivery

Prerequisites and Co-requisites

Recomended Optional Programme Components

Course Contents

Recomended or Required Reading

Planned Learning Activities and Teaching Methods

Assessment Methods

Further Notes About Assessment Methods

Assessment Criteria

Language of Instruction

Course Policies and Rules

Contact Details for the Lecturer(s)

Office Hours

Work Placement(s)

Workload Calculation

Contribution of Learning Outcomes to Programme Outcomes

CONTACT INFORMATION