Determinants of pseudo-laplacians and ^ ({ reg) } (1) for spinor bundles over Riemann surfaces

Let P be a point of a Riemann surface X. We study self-adjoint extensions of Dolbeault Laplacians in hermitian line bundles L over X initially defined on sections with compact supports in X\P\. We define the -regularized determinants for these operators and derive the comparison formulas for them. We introduce the notion of the Robin mass of L. This quantity enters the comparison formulas for determinants and is related to the regularized (1) for the Dolbeault Laplacian. For spinor bundles of even characteristic, we find the explicit expressions for the Robin mass.