In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with an arbitrary quasi-permutation monodromy group. The solution is given in terms of Szegö kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and de...

For a one-dimensional family of abelian varieties equipped with principal theta divisors a canonical limit is constructed as a pair consisting of a reduced projective variety and a Cartier divisor on it. Properties of such pairs are established.

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of vector bundles on an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit...

The object of this paper is the theta divisor of the compactified Jacobian of a nodal curve. We determine its irreducible components and give it a geometric interpretation. A characterization of hyperelliptic irreducible stable curves is appended as an application.

In this paper we study an invariant for oriented three-manifolds with b1 > 0, which is defined using Heegaard splittings and the theta divisor of a Riemann surface. The paper is divided into two parts, the first of which gives the definition of the invariant, and the second of which identifies it with more classical (torsion) invariants of three-manifolds. Its close relationship with Seiberg-Wi...

We classify GV -subschemes of five-dimensional ppavs, proving the main conjecture in [PP08b] in this case. This result is implied by a more general statement about subvarieties of minimal cohomology class whose sum is a theta divisor.

where (Pη,Ξη) is the principally polarised Prym variety corresponding to a 2-torsion point η ∈ J2(C), or is the Jacobian (J(C),Θ) in case η = 0; and where H ± denotes even/odd sections. The interest of (0.1) is that it is one step in the direction, initiated by Beauville and others ([B1], [B2], [BNR]), of relating the Verlinde spaces of C—i.e. the vector spaces H(M,L) where M is some moduli spa...

We construct maps from moduli spaces of vector bundles on a Riemann surface X to opers on X, using nonabelian theta functions. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle...