Finite Generation of Canonical Ring by Analytic Method

TL;DR

This paper presents an analytic proof demonstrating the finite generation of the canonical ring for complex algebraic manifolds of general type, addressing challenges of infinite blow-ups in the traditional approach.

Contribution

It introduces an analytic method that effectively manages the infinite blow-up problem in proving finite generation of the canonical ring.

Findings

Provides an overview of the analytic proof technique.

Shows how the method handles infinite blow-ups.

Contributes to the understanding of algebraic geometry via analytic methods.

Abstract

In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425).