Let \(X_ 0\) be a compact Riemannian surface of genus \(g>1\) with local coordinate n, on which a family of Riemannian metrics \(ds^ 2(\varepsilon)\) depending on a harmonic Beltrami differential are given. To each such metric a Riemannian surface \(X_{\varepsilon}\) is associated. Denote by \(\Delta_ n(\epsilon)\) the corresponding Laplace operator (acting on sections of an appropriate complex line bundle). The purpose of the paper is to derive formulas for the first and second variation of the function \(\epsilon\) \(\mapsto \log \det \Delta_ n(\varepsilon)\). The formula for the second variation turns out to be a local version of the Atiyah–Singer index formula.