I.The Basic Information of the Course(Time New Rome, 12 points, bold font)
Course Number:202012420030
The EnglishName of theCourse：Algebraic Geometry
The ChineseName of theCourse：代数几何
In-classHours andAllocation:Total class hours:32, classroom teaching: 32
Credit(s):2
Semester:2
AppliedDiscipline(Professional Degree Category)：Master/Doctor in Pure Mathematics
CourseObjectOriented：academicMaster/Doctor
EvaluationMode:Process Evaluation
TeachingMethod:Seminar-style Teaching
CourseOpening Department:
Notes:Filling explanation to applied discipline: the general courses like foreign languages, ideological and political theory course may fill in “AllDisciplines”,Mathematics general course may fill inScience andEngineeringDiscipline, and the other courses fill in “Name ofApplicableDiscipline” according to actual teaching objects, and the number of applicable discipline may be more than one.
II.Prerequisite Course( Time New Rome 12 points bold font)
Topology, Abstract Algebras, Complex Analysis
III.The Objectives and Requirements of the Course(Time New Rome 12 points, total words: about 200 words)
The course is a basic course for graduate students for Master/Doctor degree in Pure mathematics, especially for . The aim of the course is to help students to grasp the basic ideas, theories and methods in Algebraic Geometry and have the ability to apply the knowledge of Algebraic Geometry to solve problems in mathematics.
Algebraic Geometry relates algebra, geometry and Number Theory. We hope student to enlarge their mathematical sight by taking the course.By learning this course, the students should reach the following basic requirements: get familiar with some basic Algebraic geometric terminologies,concepts , ideals; understood the current developments in Algebraic Geometry.
IV. The Content of the Course(Time New Rome, 12 points, bold font, 1000-2000 words)
Description:
Chapter 1 A brief introduction to Algebraic Geometry (12classhours)
1.1Definition of algebraic varieties and homomorphic morphisms
1.2 Basic properties of varieties:dimension/ normality/ smoothness/ compeletion
Chapter 2Complexmanifolds(12classhours)
2.1Complex structures and Manifolds
2.2 Cohomologies of Holomorphic Vector Bundles
2.3Hizebruch-Riemannian Roch Theorem andKodaria Vanishing Theorem
Chapter 3Kahler Geometry(4classhours)
3.1 Holomorphic tensor and sectionalcurvature
3.2 Frankel conjecture and Kahler Einstein manifolds
Chapter4Kodaira-Kuranishdeformation of complex structure (4classhours)
4.1Kodariadeformation andMaurer-Cartanequations
Political education；
1，Chern，shiing-shen, was an Wolf medalist Chinese mathematician. He was born
in war time China in early twentieth century. Even though he made great contributions to
differential geometry. We should learn his patriotism and enthusiasm for mathematics.
2, Gang Xiao, was a Chinese mathematician. He made a great contribution to classification of algebraic surfaces. He also bring many innovations in Online teaching and Solar energy equipment. We should learn his patriotism and efforts to transform science and theory into technology.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1.陈省身 微分几何北京大学出版社2000
2.S．Donaldson, P Kronheimer Geometry of four Manifolds, Oxford University Press (1999)
3. R.Hartshorne, Algebraic Geometry, Springer-Verlag,Berlin,1999.
4. Daniel Huybrechts. Complex Geometry, An Introduction.世界图书出版社, 2000。
5.S.Iitaka, Algebraic Geometry, Springer-Verlag,Berlin,2002.
6.D.Mumford, Algebraic Geometry I. Complex Projective varieties,Springer-Verlag,Berlin,1976.
7. D. Mumford. The Red Book of Varieties and Schemes. Lecture Notes in Mathematics 1358. Springer- Verlag,2008.
8.李克正代数几何初步,大学数学科学丛书2,科学出版社，2004
9. J. P. Serre. Faisceaux algébriques cohérents. Ann. of Math,2005,61: 197-278.
10.I.R. Shafarevich, basic Algebraic Geometry, Grundlehren der MatheMatischen Wissenschaften, 213,Springer Verlag, Berlin, 1974.
B. Learning Resources (Time New Rome 12 points)
1.http://www.arxiv.org
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