# 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)

`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 **C**^{N} for some N, thus we obtain a new meromorphic connection on **C**^{N}. The connection of this type is a deformation of a connection on **C**^{N} 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)-curves^{1}, 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)∈ **C**^{2+d} | y^{n}-x^{s}-∑ λ_{q(i,j)} x^{i}y^{j}=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.