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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Honda Nobuhiro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Fri5-6(H119A)
- Group
- -
- Course number
- MTH.B506
- Credits
- 1
- Academic year
- 2017
- Offered quarter
- 2Q
- Syllabus updated
- 2017/3/17
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
We explain basic materials in the theory of complex manifolds. We introduce basic examples of complex manifolds, and then introduce the concept of harmonic differential forms. These establish basic results in geometry. For complex manifolds, the concept of Dolbeault groups naturally makes sense, and we can represent their elements by harmonic forms. If the manifolds are moreover Kaehler, harmonic forms as a real manifold and a complex manifold agree, and it induces a constraint on the topology of such manifolds.
Student learning outcomes
・to be familiar with basic examples of complex manifolds and basic concept such as first Chern class and blow-up,
・to understand cohomology and harmonic forms
・to know (Hodge) decomposition of differential forms on a closed real manifold and compact complex manifolds
Keywords
complex manifold, complex projective variety,
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | holomorphic functions, complex manifolds, complex projective space, algebraic variety | Details will be provided during each class session. |
| Class 2 | Weierstrass preparation theorem, intersection of analytic cylcles | |
| Class 3 | harmonic forms and the Hodge decomposition of differential forms | |
| Class 4 | harmonic forms and Hodge decomposition (the case of complex manifolds) | |
| Class 5 | Dolbeault cohomology, Dolbeault's theorem, Hodge decomposition on compact Kaehler manifolds | |
| Class 6 | holomorphic line bundle, chern class, positivity, vanishing theorem | |
| Class 7 | computations of Dolbeault cohomology | |
| Class 8 | blowup and blow down, birational transformation |
Textbook(s)
none
Reference books, course materials, etc.
P. Griffiths, J. Harris, "Principles of Algebraic Geometry", Wiley-Interscience
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.B505 ： Advanced topics in Geometry E1
- MTH.E532 ： Special lectures on advanced topics in Mathematics H
- MTH.C301 ： Complex Analysis I
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basic knowledge on geometry and complex analysis is required. MTH.B505 is assumed to be taken.
