  We discuss an analogue of Riemann-Roch theorem for curves with an infinite
number of handles. We represent such a curve X by its Shottki model, which is
an open subset U of CP^{1} with infinite union of circles as a boundary. An
appropriate bundle on X is \omega^{1/2} \otimes L, L being a bundle with (say)
constants as gluing conditions on the circles. An admissible section of an
appropriate bundle on X is a holomorphic half-form on U with given gluing
conditions and H^{1/2}-smoothness condition.
  We study the restrictions on the mutual position of the circles and the
gluing constants which guarantee the finite dimension of the space of
appropriate sections of admissible bundles, and make the Riemann-Roch theorem
hold. The resulting Jacobian variety is described as an infinite-dimension
analogue of a torus.
