The Miyaoka-Yau Type Inequality for Complete Intersection Subvarieties in a Product of Projective Spaces

Professor Introduction

H. S | Ph.D. in Mathematics

Home Institute：Shanghai Normal University

[ Research Interests ] Algebraic geometry, particularly the construction of Bridgeland stability conditions, geometric invariant theory, and the classification of algebraic varieties
[ Teaching Experience ] Associate Professor, Department of Mathematics, Shanghai Normal University
[ Publications ] Over ten papers in high-profile journals such as the Journal of Differential Geometry, Advances in Mathematics,International Mathematics Research Notices, and Transactions of the American Mathematical Society

Project Description

This project delves into the Miyaoka-Yau-type inequalities for complete intersection subvarieties in products of projective spaces. The Miyaoka-Yau inequality, discovered by Shigefumi Mori and Shing-Tung Yau, is a significant result in algebraic and differential geometry concerning the Chern classes (or characteristic classes) of algebraic surfaces. While this inequality has been extended to higher dimensions, the complexity of Chern classes in these cases presents additional challenges. This project focuses on calculating the characteristic classes of complete intersection subvarieties within products of projective spaces and establishing inequalities between these classes, thereby deriving Miyaoka-Yau-type inequalities for such varieties.

Project Keywords

Project Outline

Part 1 : Introduction to Projective Spaces and Complete Intersections
• Fundamentals of projective spaces
• Definition and properties of complete intersections
• Examples and basic constructions

Part 2 : Chern Classes and Characteristic Classes
• Introduction to Chern classes and their significance in algebraic geometry
• Calculation of Chern classes for various algebraic varieties
• Characteristic classes in the context of complete intersections

Part 3 : Miyaoka-Yau Inequality and Its Extensions
• Detailed study of the Miyaoka-Yau inequality for algebraic surfaces
• Extensions of the Miyaoka-Yau inequality to higher dimensions
• Challenges and complexities in higher-dimensional cases

Part 4 : Calculating Chern Classes in Products of Projective Spaces
• Methods for calculating Chern classes in products of projective spaces
• Techniques for dealing with complete intersection subvarieties
• Practical examples and problem-solving sessions

Part 5 :  Establishing Miyaoka-Yau Type Inequalities
• Developing inequalities between Chern classes for complete intersections
• Deriving Miyaoka-Yau type inequalities for subvarieties in products of projective spaces
• Advanced topics and open problems in the field

Suitable for

• High School Students preparing for research in algebraic geometry and related fields
• Graduate students and advanced undergraduates in mathematics, particularly those with an interest in algebraic geometry and differential geometry