Complex curves of genus three, Kummer surfaces and Quillen metrics (Q2573736)

Let \(C\) be a smooth, complex projective curve of genus 3 and \( \lambda(O_C)\) the determinant of the cohomologies of \(C\). The authors define for a pair \((u, \tau) \in (\mathbb{C}^4-0 \times H_2)\), where \(H_2\) is the Siegel upper half-space, whose existence depends on a choice of the double covering \(\tilde{C} \rightarrow C\), a non-zero element \(\varphi(u, \tau) \in \lambda (O_C)\) . Denoting by \(\| \cdot \|\) the Quillen metric on \( \lambda_C\) with respect to \(k_C\), the authors give an explicit formula for \(\| \varphi(u, \tau) \|^2\) and state it as main theorem 0.1. Their main theorem is a generalization of the results presented for example by \textit{D. B. Ray} and \textit{I. M. Singer} [Ann. Math. (2), 98, 154--177 (1973; Zbl 0267.32014)], \textit{G. Faltings} [Ann. Math. 119, 387--424 (1984; Zbl 0559.14005)] and more recently by \textit{K. Yoshikawa} [J. Differ. Geom. 52, 73--115 (1999; Zbl 1033.58029)]. The paper is organized as follows. In section 1, for a pair \((F, h_F)\) of a hermitian vector bundle on \(X\), for \(X\) a compact Kähler manifold with Kähler metric \(k_X\), they define the Ray-Singer analytic torsion with respect to \(k_X\) in definition 1.1. In definition 1.2 the equivariant Quillen metric is defined. In section 2.3.1 they compute the \(\mu_2\) equivariant Quillen metric on \( \lambda_{\mu_2} (O_{A_{\tau}})\) for \(A_{\tau}\) an abelian surface and on \( \lambda_{\mu_2}({L_{\tau}}^{-2})\). In subsection 2.3.2 they continue with the computation of the Quillen metric on \(\lambda_{\mu_2}( L_{\tau}^2 \otimes K_{A_{\tau}})\). In section 3 subsection 3.1 Kummer's quartic surface \(R_{\tau}\) is defined. In subsection 3.2, theorem 3.2 they prove that \(R_{\tau}\) is self-dual. In section 4, proposition 4.2 the authors show that every smooth projective curve of genus 3 is isomorphic to a divisor of a Kummer surface constructed in as in section 3. In subsection 5.2 of section 5 they state the main result of the paper as main theorem 5.3 and prove it. The paper ends in section 6 where they prove a technical result stated as proposition 6.1 to prove lemma 3.7 which is used to prove the exact duality of Kummer's quartic surface (theorem 3.2).