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An analytic proof of the stable reduction theorem
Jian Song1, Jacob Sturm2, Xiaowei Wang2
1Department of Mathematics, Rutgers University, Piscataway, NJ 08854 USA.
Mathematische Annalen
|November 15, 2024
Summary
The stable reduction theorem is extended to algebraic curves over the rational numbers. This is achieved by using Kähler-Einstein metrics to find the limiting stable curves at punctures.
Area of Science:
- Algebraic Geometry
- Differential Geometry
Background:
- The stable reduction theorem is a fundamental result in algebraic geometry.
- It concerns the completion of families of curves over punctured curves by adding stable curves.
Purpose of the Study:
- To provide a new proof of the stable reduction theorem for curves defined over the rational numbers.
- To utilize Kähler-Einstein metrics to identify the limiting stable curves.
Main Methods:
- Applying the theory of Kähler-Einstein metrics to the fibers of the curve family.
- Developing techniques to obtain limiting stable curves at punctures.
Main Results:
- A novel proof of the stable reduction theorem for curves over the rational numbers.
- Identification of specific stable curves that complete the family.
Conclusions:
- The study successfully extends the stable reduction theorem to a new domain.
- The use of Kähler-Einstein metrics offers a powerful approach to problems in algebraic geometry.
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