Analogues of the Cauchy kernel on a Riemann surface and some of their applications

Under certain restrictions on the behavior in the neighborhood of the ideal boundary analogues of the Cauchy kernel corresponding to an arbitrarily given finite divisor are constructed on an arbitrary Riemann surface. As a first application of these kernels a new proof of the Riemann–Roch theorem is given that differs from the well-known ones (even in the case of a ompact Riemann surface) by its simplicity and constructivity (bases of the corresponding spaces of functions and differentials are explicitly expressed in terms of elementary Abelian differentials of the first and third kinds). Using integrals of Cauchy type with kernels as mentioned above the author gives an explicit solution of the “jump” problem for piecewise meromorphic functions as well as for differentials. In addition, the results are applied to the study of the Riemann and Markushevich boundary value problems. Normal solvability of these problems is proved and their indices are computed.
Bibliography: 26 titles.