Dmitry Leykin (MIMS and Institute of Magnetism, Kiev)
Dmitry.Leykin@manchester.ac.uk Flat meromorphic connections are an object of interrelated research in Physics, Geometry, and Singularity Theory (first to mention: the KZB and WDVV equations, Frobenius structures on manifolds, and Saito structures).
The talk is devoted to the construction, recently found by Victor Buchstaber and the speaker, of a flat connection on the universal bundle of Jacobians of a plane algebraic curve. The space of such a bundle is birationally equivalent to CN for some N, thus we obtain a new meromorphic connection on CN. The connection of this type is a deformation of a connection on CN with only simple rational singularities along a certain prescribed arrangement of rational hypersurfaces. The parameters of deformation are parameters of the underlying family of curves.
As the starting point we take the families of (n,s)-curves1, for which an effective theory of sigma-function is developed. Using heat-type differential operators that annihilate sigma-function, we construct explicitly the Lie algebra of derivations of the field of Abelian functions on the Jacobian of the curve. The derivations define a flat connection on the universal bundle of Jacobians.
The talk is intended to be accessible for a broad audience.
V={(x,y; λ)∈ C2+d | yn-xs-∑ λq(i,j) xiyj=0 },
(summation over i,j≥ 0, q(i,j)>0), where q(i,j)=(n-j)(s-i)-ij and d=n s-(n-1)(s-1)/2. Generic curve from the family has genus g=(n-1)(s-1)/2. It is known that an arbitrary plane curve has an (n,s)-model.