Manchester Geometry Seminar 2006/2007

30 November 2006. Room MSS/C013A, Mathematics and Social Sciences building, Sackville Street. 2.30pm

A Canonical Flat Meromorphic Connection on the Universal Bundle of Jacobians

Dmitry Leykin (MIMS and Institute of Magnetism, Kiev)

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.

1 Fix a pair of co-prime numbers (n,s). Then the (n,s)-family of plane algebraic curves is the set

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.