Balanced metrics and Chow stability of projective bundles over Riemann surfaces (Q2839368)

In the previous paper [Duke Math. J. 153, 573--605 (2010; Zbl 1204.32013)] the author proved the following: NEWLINENEWLINENEWLINENEWLINE Let \(X\) be a compact complex manifold, with discrete \(\text{Aut}(X)\) and with a Kähler metric of constant scalar curvature, whose Kähler form is in \(2 \pi c_1(L)\) for some ample line bundle \(\pi: L \longrightarrow X\). If \(\pi^E: E \longrightarrow X\) is a Mumford-stable holomorphic vector bundle of rank \(r\), then there exists \(k_0\) such that \((\mathbb P E^*, \mathcal O_{\mathbb P E^*}(1) \otimes \pi^* L^k)\) is Chow-stable for any \(k \geq k_0\).NEWLINENEWLINENEWLINENEWLINE By the results of Luo, Phong, Sturm, Wang and Zhang, the theorem was obtained as a corollary of the proof that \((\mathbb P E^*, \mathcal O_{\mathbb P E^*}(1) \otimes \pi^* L^k)\) admits a balanced metric for \(k > > 0\). In this paper, the author gives a new and simpler proof of this fact when \(X\) is a compact Riemann surface of genus \(g \geq 2\). Indeed, in this case it is proved that, for any positive integer \(d\), \((\mathbb P E^*, \mathcal O_{\mathbb P E^*}(d) \otimes \pi^* L^k)\) admits a balanced metric for any \(k > > 0\).