A geometric approach to problems in birational geometry
Journal
Proceedings of the National Academy of Sciences of the United States of America
Journal Volume
105
Journal Issue
48
Pages
18696-18701
Date Issued
2008
Author(s)
Yau, Shing-Tung
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 that 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. © 2008 by The National Academy of Sciences of the USA.
Subjects
Classification; Pluricanonical systems; Pseudonorms; Torelli-type theorems; Varieties of general type
Other Subjects
article; geometry; isometrics; mathematics; priority journal; spatial discrimination; Algorithms; Mathematics; Models, Statistical
Type
journal article