Log Picard Algebroids and Meromorphic Line Bundles
| Autor: | Marco Gualtieri, Kevin Luk |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
Lie algebroid Pure mathematics General Mathematics 010102 general mathematics Divisor (algebraic geometry) 01 natural sciences Mathematics - Algebraic Geometry Mathematics::Algebraic Geometry Differential Geometry (math.DG) Line bundle 0103 physical sciences FOS: Mathematics De Rham cohomology 010307 mathematical physics Projective plane 0101 mathematics Algebraic Geometry (math.AG) Mathematics::Symplectic Geometry Hodge structure Projective variety Mathematics Meromorphic function |
| Zdroj: | International Mathematics Research Notices. 2021:16592-16635 |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rnz272 |
| Popis: | We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibres degenerate along the divisor. We give a geometric description of such bundles and establish a classification theorem for them, showing that they correspond to a subgroup of the holomorphic line bundles on the divisor complement. Importantly, these holomorphic line bundles need not be algebraic. Finally, we provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane. 29 pages |
| Databáze: | OpenAIRE |
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