Special lectures on advanced topics in Mathematics B

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Makoto Enokizono / Shou Yoshikawa

Class Format

Lecture

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Class

-

Course Code

MTH.E432

Number of credits

200

Course offered

2026

Offered quarter

3Q

Syllabus updated

Mar 5, 2026

Language

Japanese

Syllabus

Course overview and goals

"The main theme of this intensive course is the minimal model theory for projective morphisms between complex analytic spaces and stacks. We begin by reviewing the basic notions of complex analytic spaces and algebraic geometry, and then study the fundamental concepts of minimal model theory. Finally, we explain the minimal model program and learn that it can be carried out in a gluable manner.
Minimal model theory is a fundamental language in algebraic geometry, particularly in birational geometry, and has a wide range of applications. At the same time, it is a highly general theory and can be difficult for beginners to grasp. In this course, we emphasize that minimal model theory can be viewed as a natural extension of the classical theory of linear systems in algebraic geometry. We also discuss the aspect that, in the relative setting of projective morphisms between complex analytic spaces, the minimal model program runs in a (certain sense) functorial way."

Course description and aims

"・To understand the basic concepts of complex analytic spaces and algebraic varieties.
・To understand the fundamental definitions and concepts of minimal model theory.
・To understand the existence (or termination) of the minimal model program.
・To understand the gluing property of the minimal model program."

Keywords

Complex manifold, Complex analytic space, Minimal model theory, Birational geometry

Competencies

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

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Objectives

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to course material.

Textbook(s)

None in particular.

Reference books, course materials, etc.

Details will be provided during each class session.

Evaluation methods and criteria

Assignments (100%)

Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV
• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II

Prerequisites

Basic knowledge on algebra is expected.