2017 Advanced topics in Geometry E1

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Academic unit or major
Graduate major in Mathematics
Honda Nobuhiro 
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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


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



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.

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