A new geometric approach to problems in birational geometry

Details A new geometric approach to problems in birational geometry A new geometric approach to problems in birational geometry

2008-11-18

arxiv.org/abs/0811.2965v1

Mathematics

Algebraic Geometry

13 pages, to appear in PNAS

Scientific paper

10.1073/pnas.0809030105

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.

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