Log Picard algebroids and meromorphic line bundles

TL;DR

This paper introduces logarithmic Picard algebroids and meromorphic line bundles on smooth projective varieties with divisors, classifies them via mixed Hodge structures, and provides explicit construction methods, extending the theory of line bundles to degenerating fibers.

Contribution

It defines and classifies logarithmic Picard algebroids and meromorphic line bundles, linking them to mixed Hodge structures and extending line bundle theory to divisors.

Findings

Log Picard algebroids are classified by a subspace of de Rham cohomology.

Meromorphic line bundles generalize line bundles with fibers degenerating along divisors.

Explicit constructions of meromorphic line bundles are provided for specific divisors.

Abstract

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.