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Determinants of pseudo-laplacians and $\zeta^{({\rm reg})}(1)$ 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 $X\backslash\{P\}$. We define the $\zeta$-regularized determinants for these operators and derive the comparison formulas for them. We introduce …