2016 Advanced topics in Geometry H

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Academic unit or major
Graduate major in Mathematics
Honda Nobuhiro  Tanabe Masaharu 
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Course description and aims

This course covers the geometry of Riemann surfaces, in particular the theories on intersection points of regular mapping between compact Riemann surfaces, including the newest material. As the lecture progresses, the instructor will deal with differential forms, Hodge decomposition, Jacobian varieties, and the Lefschetz trace formula. This course is a continuation of Advanced Geometry 4 (Special Lectures on Geometry IV) offered in 2014, but fundamentals will be covered again, so students will be able to follow the lectures even if they take this course alone.
Students will gain an understanding of the newest topics in this field. To solidify understanding, exercise problems will be given out during lecture, for students to submit as a report.

Student learning outcomes

To determine the existence and estimate the number of solutions of a given equation is one of the fundamental problems in mathematics. To estimate the number of coincidences of morphisms between manifolds is a version of manifolds of this problem.
This course deals with Riemann surfaces which are of minimal dimension as complex manifolds. Several simple theorems are given by virtue of the low dimension.
By completing this course, students will be able to understand the latest topics in this field.


Riemann surfaces, differential forms, Hodge decomposition, Jacobian varieties, the Lefschetz trace formula.

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Each class is devoted to fundamentals. To allow students to get deep understanding of the course contents, problems related to the contents of this course are provided. Students should submit reports solving them.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Riemann surfaces Including the second class and subsequent, students should understand each topics in depth.
Class 2 Holomorphic maps between Riemann surfaces
Class 3 differential forms, Hodge decomposition
Class 4 Jacobian varieties
Class 5 fixed points, coincidences
Class 6 the Lefschetz trace formula
Class 7 Holomorphic Lefschetz number
Class 8 the Eichler trace formula


H. M. Farkas and I. Kra, Riemann surfaces, GTM 71, Springer-Verlag

Reference books, course materials, etc.

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley classic library ed., John Wiley & Sons, Inc.

Assessment criteria and methods


Related courses

  • MTH.C301 : Complex Analysis I
  • MTH.C302 : Complex Analysis II
  • MTH.B301 : Geometry I
  • MTH.B302 : Geometry II
  • MTH.B341 : Topology

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


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