3. 강의목표
Abelian varieties are one of fundamental objects studied in algebraic geometry and number theory.
There are many deep and interesting results established and while still many problems to be investigated.
The aim of this course is to introduce abelian varieties from mainly the analytic theory and to provide working knowledge for research papers.
4. 강의선수/수강필수사항
선수과목 (권장)
- MATH 502 Algebra II
- MATH 510 Complex Analysis
- MATH 508 Introduction to Algebraic Geometry
7. 참고문헌 및 자료
References:
[1] David Mumford, Abelian varieties. Chap. 1
[2] Olivier Debarre, Complex tori and abelian varieties.
[3] P. Griffiths and J. Harris, Principles of Algebraic Geometry. Chap. 2 Sections 2 and 6.
[4] Ching-Li Chai, The period matrices and theta functions of Riemmann. Available in his homepage
9. 수업운영
Topics:
Elliptic curves and modular curves, complex tori, line bundles, classification of line bundles on complex tori, cohomology groups of complex tori and abelian varieties, ample and very ample line bundles and projective morphism, the Riemann-Lefschetz theorem, polarizations and dual abelian varieties, construction of abelian varieties: moduli and CM abelian varieties, Riemann surfaces and Jacobians, Riemann bilinear relations, endomorphism algebras of abelian varieties, moduli spaces of abelian varieties. If time permits, abelian varieties over finite fields: endomorphisms, the Honda-Tate theorem and polarizations.
Prerequisite: Graduate Algebra, Complex analysis. Notions of complex manifolds, algebraic varieties/schemes, and sheaf cohomology will be helpful.
This course consists of lectures and students’participation.
Students’participation include: attendance and interaction in class, provide (more detailed) proofs of some results whose proofs missed (or sketched) in lectures, cooperative work for typing lecture notes, and possible presentation (to be determined later, may only for a limited number of graduate students).
10. 학습법 소개 및 기타사항
*** 수학과 안내 사항 ***
- 강의교수: 2024-1학기 방문 예정인 "Chia-Fu Yu" 방문 교수
- University of Pennsylvania에서 정수론을 전공한 Taiwan을 대표하는 정수론자 임.
- 정수론의 핵심 주제 중의 하나인 Shimura variety의 expert 로 특강 예정임
11. 장애학생에 대한 학습지원 사항
- 수강 관련: 문자 통역(청각), 교과목 보조(발달), 노트필기(전 유형) 등
- 시험 관련: 시험시간 연장(필요시 전 유형), 시험지 확대 복사(시각) 등
- 기타 추가 요청사항 발생 시 장애학생지원센터(279-2434)로 요청