Classification of Lagrangian fibrations

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Hyperkahler manifolds arise in many different areas of high energy theoretical physics, for example as spaces parametrizing Yang-Mills instantons, magnetic monopoles, and Higgs fields. In their algebro-geometric manifestation, hyperkahler manifolds appear as holomorphic symplectic manifolds. Fibrations on holomorphic symplectic manifolds are known as Lagrangian fibrations, since the fibers must be Lagrangian with respect to the holomorphic symplectic structure. In addition, the generic fibres must be abelian varieties, and Lagrangian fibrations may be viewed as higher-dimensional analogues of elliptic K3 surfaces. The Hitchin system is a celebrated non-compact example. In this project the P.I. will study the classification of (compact) Lagrangian fibrations. The goals are to classify Lagrangian fibrations in some particular cases: when the fibers are Jacobians of curves or principally polarized, when the fibration is locally isotrivial, in dimension four (abelian surfaces as fibers). The P.I. will also explore new constructions of Lagrangian fibrations, and address the question of finiteness of deformation classes.This project will develop some particular kinds of geometry that are known to arise in theoretical physics, particularly in quantum field theory and string theory. The main focus will be on hyperkahler geometry, a class of spaces admitting Ricci-flat metrics, i.e., solutions to Einstein's equations in a vacuum. The P.I. will also study generalized complex geometry, which was invented by mathematicians to bridge complex and symplectic geometry, on which type IIA and IIB string theory are based, and to help unravel the Mirror Symmetry phenomenon predicted by string theorists. In short, these are all innovative new kinds of geometry developed largely in response to the demands of theoretical physicists, but also of inherent mathematical interest. The P.I. will actively promote the exchange of ideas between mathematicians and physicists by organizing and participating in cross-disciplinary conferences and workshops, and is also involved in the training of graduate students in these topics.