Notes on Yoshida's coordinates on Hitchin's Prym cover (Q2390671)

The paper under review contains an expository account of a certain atlas of coordinate systems on a Prym variety that covers the moduli space of semistable vector bundles of rank \(2\) and trivial determinant on a Riemann surface. These coordinate systems arose originally in the work of \textit{T. Yoshida} [Ann. Math. (2) 164, No. 1, 1--49 (2006; Zbl 1112.14040)]. Let \(\Sigma\) be a compact Riemann surface of genus \(g\geq 2\), and let \(K\) denote its canonical bundle. Let \(q\in H^0(\Sigma,K^2)\) be a quadratic differential on \(\Sigma\) with simple zeros. Then, the subvariety \(\widetilde{\Sigma}\) consisting of all \(a_x\in K_x\) (\(x\in \Sigma\)) such that \(a_x^2 = q(x)\) is a double covering \(pr: \widetilde{\Sigma}\rightarrow \Sigma\) simply branched over the zeros of \(q\). The Prym variety in question is the subvariety \(Prym(\widetilde{\Sigma})\) of the Jacobian \(J^{2g-2}(\widetilde{\Sigma})\), which consists of line bundles \(L\) of degree \(2g-2\) on \(\widetilde{\Sigma}\) such that \(pr_*(L)\) has trivial determinant. The construction of the coordinate systems on \(Prym(\widetilde{\Sigma})\) involves two other auxiliary abelian varieties, and is done using the combinatorial topology of the Riemann surface \(\Sigma\).