A Certain Type of I R R E G U L a R Algebraic Surfaces

A. Van de Ven has pointed out in connection with his recent results(1) that there are not many known examples of compact connected complex 4-manifolds with positive indices. The purpose of this note is to exhibit a series of compact connected complex 4-manfolds M . . . . n, m = 2, 3, 4, ..., with positive indices. Each complex 4-manifold Mn.m is an irregular algebraic surface having a structure of locally non-trivial complex analytic family of non-singular algebraic curves of genus m(2n1) whose base space is a compact Riemann surface. The complex 4-manifold Mn,m serves as an example of differentiable fibre bundle such that the index of the bundle space is not equal to the product of the indices of the fibre and of the base space(2). Let Ro be a compact Riemann surface of genus n > 2. The surface R o is topologically a "sphere with n handles". Letting Yo be a "mer id i an" circle on a handle of Ro, we construct a 2-sheeted unramified covering surface R of Ro with a covering map 2:R--} R o such that R 2-1(~o) consists of two