Balanced metrics and Chow stability of projective bundles over Kähler manifolds
The paper is devoted to the proof of the following: 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 \(\pi: L \to X\). If \(\pi^E: E \to X\) is a Mumford-stable holomorphic vector bundle of rank \(r\), then there exists \(k_o\) such that \((\mathbb P E^*, \mathcal O_{\mathbb P E^*}(1) \otimes \pi^* L^k)\) is Chow-stable for any \(k \geq k_o\). The result is obtained by proving that if \(E\) is Mumford-stable, then \((\mathbb P E^*, \mathcal O_{\mathbb P E^*}(1) \otimes \pi^* L^k)\) admits a balanced metric for \(k > > 0\). This is equivalent to Chow-stability by the results of Zhang, Luo and Phong and Sturm [\textit{S. Zhang}, Compositio Math. 104, No. 1, 77--105 (1996; Zbl 0924.11055); \textit{H. Luo}, J. Differ. Geom. 49, No. 3, 577--599 (1998; Zbl 1006.32022); \textit{D. H. Phong} and \textit{J. Sturm}, Commun. Anal. Geom. 11, No. 3, 565--597 (2003; Zbl 1098.32012)]. The proof is based on ideas of [\textit{S. K. Donaldson}, J. Differ. Geom. 59, No. 3, 479--522 (2001; Zbl 1052.32017)]. It consists in showing at first the existence of an almost balanced metric, determined by an asymptotic expansion of the Bergman kernel, and secondly that this metric can be modified into a balanced one.
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