Degeneration of the Mukai system

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The background to this research proposal is the existence of naturally occurring integrable systems in algebraic geometry. Traditionally, such a system is a system of differential equations which has enough constants of the motion to allow one in principle to solve completely the equation. Geometrically these constants of the motion are functions on the phase space which Poisson commute and whose common level sets have the structure of a torus. The classical example is the motion of a pendulum which can be solved with elliptic functions; the torus is the curve or more precisely its Picard variety. Another, more complicated one, the geodesics on an ellipsoid, requires hyperelliptic functions, where the torus is a higher-dimensional abelian variety. More generally, one can adopt a coordinate-free viewpoint and consider algebraic varieties which are symplectic and have a fibration by abelian varieties, but they are not so easy to find. There are two quite broad families of such algebraic varieties: one has come to be known as the Hitchin system and the other the Mukai system.The Hitchin system arises from the study of moduli space of Higgs bundles on an algebraic curve. The data consists of a curve and a simple Lie group, and sometimes extra information concerned with marked points on the curve. The Mukai system requires a curve in a K3 surface and a simple Lie group, though most work involves linear groups. Donagi, Ein and Lazarsfeld showed that the Mukai system can be regarded as a nonlinear deformation of the Hitchin system. K3 surfaces have been studied intensely for many years, especially by the proposed visitor, and they have an internal geometry much richer than that of a single curve lying on them. One can therefore expect new features to appear in the Mukai moduli space.The Hitchin system has recently become a valuable tool in other areas of mathematics such as number theory and representation theory. Furthermore, thanks to the work of physicists Witten and Kapustin relating its special properties to electric-magnetic duality, a number of new viewpoints and results have been produced, connecting in particular to the Langlands programme, a unifying vision of many mathematical entities which originates in number theory. The proposal consists of attempting to understand the role of these new points of view in the Mukai system: looking for new structures which in the limit of the degeneration become the known ones on the Hitchin system. The structures include cohomology -- the study of the underlying topology of the spaces and natural representative cycles; the derived category of coherent sheaves -- a more refined way of capturing the relations between complex submanifolds and vector bundles over the space; and an investigation into the idea of replacing the symplectic structure on the K3 surface by the more general notion (still originally founded in differential equations) of a Poisson structure.