The Prym and Ganning Vector Bundles over the Teichmueller Space

We introduce and study the Prym vector bundle P of holomorphic Prym differentials and the Ganning cohomology bundle G over the Teichmueller space of compact Riemann surfaces of genus g2 and over the Torelli space of genus g2. We construct a basis of holomorphic Prym differentials on a variable compact Riemann surface which depends on the moduli of the compact Riemann surface and on the essential characters. From these bundles we compose an exact sequence of holomorphic vector bundles over the product of the Teichmueller space of genus g and a special domain in the complex manifold C ^2g/Z ^2g.     

The gunning cohomology bundle and the Torelli group

Kemerovo. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 3, pp. 198–203, May–June, 1990.     

Multiplicative Weierstrass Points and the Jacobi Variety of a Compact Riemann Surface

In the previous papers of the present author, a general theory of multiplicative Weierstrass points on compact Riemann surfaces for arbitrary characters was developed. In the present paper, some additional relations between multiplicative Weierstrass points on a compact Riemann surface for an arbitrary character and special subsets in the Jacobi variety, the canonical embedding of a compact Riemann surface into a projective space, are established. We not distinction between classical Weierstrass points and multiplicative Weierstrass points on a compact Riemann surface.     

The space of harmonic Prym differentials on a variable compact Riemann surface

The harmonic Prym differentials and their period classes play an important role in the modern theory of functions on compact Riemann surfaces [1–7]. We study the harmonic Prym bundle, whose fibers are the spaces of harmonic Prym differentials on variable compact Riemann surfaces and find its connection with Gunning’s cohomological bundle over the Teichmüller space for two important subgroups of the inessential and normalized characters on a compact Riemann surface. We study the periods of holomorphic Prym differentials for essential characters on variable compact Riemann surfaces.     

Weierstrass Multiplicative Points on a Compact Riemann Surface

We introduce Weierstrass multiplicative points and develop the theory of Weierstrass multiplicative points for multiplicative meromorphic functions and Prym differentials on a compact Riemann surface. We prove some analogs of the Weierstrass and Noether theorems on the gaps of multiplicative functions. We obtain two-sided estimates for the number of Weierstrass multiplicative points and q-points. We propose a method for studying the Weierstrass and Noether gaps and Weierstrass multiplicative points by means of filtrations in the Jacobi variety of a compact Riemann surface.     

The spaces of meromorphic Prym differentials on a finite Riemann surface

In the previous articles the second author started constructing a general theory of multiplicative functions and Prym differentials on a compact Riemann surface for arbitrary characters. Function theory on compact Riemann surfaces differs substantially from that on finite Riemann surfaces. In this article we start constructing a general function theory on variable finite Riemann surfaces for multiplicative meromorphic functions and differentials. We construct the forms of all elementary Prym differentials for arbitrary characters and find the dimensions of, and also construct explicit bases for, two important quotient spaces of Prym differentials. This yields the dimension of and a basis for the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters.     

Periods of Harmonic Prym Differentials on a Compact Riemann Surface

We study the period classes of closed, harmonic, and holomorphic Prym differentials on a compact Riemann surface of any genus g2 for arbitrary characters of its fundamental group. We prove that the harmonic Prym vector bundle of harmonic Prym differentials and the Gunning cohomology bundle are real-analytically isomorphic over the base of nontrivial normalized characters for every compact Riemann surface of genus g2.     

Prym differentials on a variable compact Riemann surface

The dimensions and bases of the spaces of meromorphic Prym differentials on a variable compact Riemann surface, as well as in the first holomorphic de Rham cohomology group of Prym differentials, are found for any characters.