A new geometric approach to problems in birational geometry

TL;DR

This paper introduces a geometric approach to birational geometry, showing that isometries of certain pseudonormed spaces of pluricanonical forms imply birational equivalences for varieties of general type.

Contribution

It establishes a Torelli-type theorem linking isometric pseudonormed spaces to birational maps between varieties of general type.

Findings

Isometries of pseudonormed spaces induce birational maps.

Positive answer for varieties of general type.

Provides a new geometric perspective in birational invariants.

Abstract

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them which induces these isometries? In this work a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.