Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, June 28, 2017
11:10 a.m.

PhD Candidate:  Kevin Luk
Co-Supervisors:  Marco Gualtieri, Lisa Jeffrey
Thesis title:  Logarithmic algebroids and line bundles and gerbes



In this thesis, we first 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 an 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 in a certain way 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 which need not be algebraic. We provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane.

We then proceed to introduce and develop the theory of logarithmic Courant algebroids and meromorphic gerbes. We show that under an integrality condition, a log Courant algebroid may be prequantized to a meromorphic gerbe with logarithmic connection. Lastly, we examine the geometry of Deligne and Deligne-Beilinson cohomology groups and demonstrate how this geometry may be exploited to give quantization results of closed holomorphic and logarithmic differential forms.

A copy of the thesis can be found here: ut-thesis-june19


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