### 2017　Advanced topics in Geometry E1

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Instructor(s)
Honda Nobuhiro
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Fri5-6(H119A)
Group
-
Course number
MTH.B505
Credits
1
2017
Offered quarter
1Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

In this lecture series we basic materials on sheaf theory, cohomology group with sheaf coefficients, and de Rham theorem, and so on. Cohomology group with sheaf coefficients are useful especially in the theory of complex manifolds. The course is followed by MTH.B506.

### Student learning outcomes

・to know the concept of sheaf and be familiar with basic examples
・to be able to handle exact sequence of cohomology groups
・to understand the mechanism of the proof of de Rham theorem

### Keywords

sheaf, presheaf, cohomology group, exact sequence, deRham theorem, intersection form, Poincare's theorem on intersection forms

### 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 sheaves, basic examples, homomorphism of sheaves Details will be provided during each class session.
Class 2 presheaves, basic examples, relation with sheaves, sheafification
Class 3 cohomology group with sheaf coefficients
Class 4 exact sequence of cohomology groups
Class 5 deRham theorem, double complex
Class 6 fine sheaf, Leray's theorem, acyclic covering
Class 7 Poincare's duality on homology groups
Class 8 dual cell decomposition, deRham group and intersection forms

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.B506 ： Advanced topics in Geometry F1
• MTH.E532 ： Special lectures on advanced topics in Mathematics H

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Basic knowledge on geometry (manifolds, differential forms, homology group) is required.