AI Article Synopsis
- A classical set of birational invariants involves the spaces of pluricanonical forms of a variety and associated subspaces, which are all birationally invariant.
- Each of these vector spaces has a specific metric structure, leading to what are called pseudonormed spaces.
- The main question addressed is whether a birational map can be constructed between two mildly singular projective varieties that have isometric pseudonormed spaces, and it's shown that this is possible for varieties of general type, similar to a Torelli-type theorem for birational equivalence.
Article 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.
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| http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2596269 | PMC |
| http://dx.doi.org/10.1073/pnas.0809030105 | DOI Listing |
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