On the Pseudonorm Project of Birational Classification of Algebraic Varieties

Abstract

The birational classification of complex varieties of general type has been one of the most crucial and challenging task in the development of algebraic geometry, since the time Kodaira accomplished his monumental work on the classification of compact complex surfaces. The current note serves as an exposition of a project towards birational classification of complex varieties of general type, which I initiated in 2008. The central role of this project is played by pseudonorms, some analytically defined norm-like functions, on the pluricanonical spaces of compact complex manifolds. We also survey on different approaches towards the birational Torelli type theorem, which is the initial step of the project, and indicate some directions for future developments.

Mathematics Subject Classification (2010). 14-02, 14C34, 32-01, 32G20.