R-Matrix Structure of Hitchin System in Tyurin Parameterization

Hitchin systems on ll-curves

Hecke-tyurin Parametrization of the Hitchin and Kzb Systems

We study the parametrization of the moduli space Bun2(C)L of rank 2 bundles over a curve C with fixed determinant, provided by Hecke modifications at fixed points of the trivial bundle. This parametrization is closely related to the Tyurin parametrization of vector bundles over curves. We use it to parametrize the Hitchin and KZB systems, as well as lifts of the Beilinson-Drinfeld D-modules. We...

Spin Canonical Invariants of 4-manifolds and Algebraic Surfaces

Donaldson’s celebrated construction of polynomial invariants for 4-manifolds (see for example [D–K]) has served as a model for an enormous range of recent results and constructions relating algebraic geometry and differential topology. The short written version of Donaldson’s talk at the 1990 Arbeitstagung contained the construction of jumping instantons, and was the starting point for our cons...

Tyurin parameters of commuting pairs and infinite dimensional Grassmann manifold

Commuting pairs of ordinary differential operators are classified by a set of algebro-geometric data called “algebraic spectral data”. These data consist of an algebraic curve (“spectral curve”) Γ with a marked point γ∞, a holomorphic vector bundle E on Γ and some additional data related to the local structure of Γ and E in a neighborhood of γ∞. If the rank r of E is greater than 1, one can use...

0 M ay 2 00 8 PRODUCTS OF JACOBIANS AS PRYM - TYURIN VARIETIES

Let X1, . . . , Xm denote smooth projective curves of genus gi ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to 1 + max i=1 gi. We show that the product JX1 × · · · × JXm of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n. This exponent is considerably smaller than the exponent of the struc...