Log Picard Algebroids and Meromorphic Line Bundles

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 fibers 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.

A Generalization of Integrality

In this paper, we explore a generalization of the notion of integrality. In particular, we study a near-integrality condition that is intermediate between the concepts of integral and almost integral. This property (referred to as the Ω-almost integral property) is a representative independent specialization of the standard notion of almost integrality. Some of the properties of this generalization are explored in this paper, and these properties are compared with the notion of pseudo-integrality introduced by Anderson, Houston, and Zafrullah. Additionally, it is shown that the Ω-almost integral property serves to characterize the survival/lying over pairs of Dobbs and Coykendall.

The Level 2 and 3 Modular Invariants for the Orthogonal Algebras

The ‘1-loop partition function’ of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of SL2(), and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra Br(1) and Dr(1) all of these at level k ≤ 3. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2—the Br(1), Dr(1) level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for B2(1) ≅ C2(1) and D7(1). The B2,3 and D7,3 exceptionals are cousins of the ε6-exceptional and ε8-exceptional, respectively, in the A-D-E classification for A1(1), while the level 2 exceptionals are related to the lattice invariants of affine u(1).