2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78031We consider the Dolbeault operator of $K^{1/2}$ -- the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups of $K^{1/2}$ 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.11 pages, Latex format, no figuresDifferential Geometry53C55, 53C15text