The balanced metrics and cscK metrics on polyball bundles

Abstract

In this paper, we use the the well-known Calabi ansatz, further generalized by Hwang–Singer and Apostolov–Calderbank–Gauduchon, to study the existence of constant scalar curvature Kähler (cscK for short) metrics and balanced metrics on certain holomorphic polyball bundles M which are locally expressed as $M=\Big \{(z_1,\ldots ,z_{{\mathcal {N}}}, u_1,\ldots ,u_{\ell })\in \prod _{j=1}^{{\mathcal {N}}}\Omega _j \times \prod _{i=1}^{\ell }{\mathbb {C}}^{r_i}: e^{\sum _{j=1}^{{\mathcal {N}}}\lambda _{ij}\phi _j(z_j)}\Vert u_i\Vert ^2<1,1\le i\le \ell \Big \}.$ Let $g_F$ be the natural Kähler metrics on M associated with the Kähler forms locally expressed as $\omega _F=\sqrt{-1}\partial \overline{\partial } \Big (\sum _{j=1}^{{\mathcal {N}}}\nu _j\phi _j(z_j) +\sum _{i=1}^{\ell }f_i(\sum _{j=1}^{{\mathcal {N}}} \lambda _{ij}\phi _j(z_j)+\log \Vert u_i\Vert ^2)\Big )$. Firstly, we obtain sufficient and necessary conditions for $g_F$ to be cscK metrics. Secondly, using this result, we obtain necessary and sufficient conditions for $mg_F$ to be balanced metrics for all sufficiently large positive integer numbers m. Finally, we obtain complete cscK metrics and balanced metrics on the polyball bundles over simply connected Riemann surfaces, or products of simply connected Riemannian surfaces and unit balls. The main contribution of this paper is the explicit construction of complete, non-compact cscK metrics and balanced metrics.