F eb 1 99 9 The Dolbeault operator on Hermitian spin surfaces

We consider the Dolbeault operator √ 2(∂ + ∂ *) of K 1 2 – the square root of the canon-ical line bundle which determines the spin structure of a compact Hermitian spin surface (M, g, J). We prove that all cohomology groups H 1 2)) vanish if the scalar curvature of g is non-negative and non-identically zero. Moreover, we estimate the first eigenvalue of the Dolbeault operator when the conformal scalar curvature k is non-negative and when k is positive. In the first case we give a complete list of limiting manifolds and in the second one we give non-Kähler examples of limiting manifolds.