A momentum construction for circle-invariant Kähler metrics

Examples of Kähler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kähler metrics of constant scalar curvature by ODE methods. One fruitful idea--the ``Calabi ansatz''--is to begin with an Hermitian line bundle $p:(L,h)\to(M,g_M)$ over a Kähler manifold, and to search for Kähler forms $\omega=p^*\omega_M+dd^c f(t)$ in some disk subbundle, where $t$ is the logarithm of the norm function and $f$ is a function of one variable.

Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kähler metric $g$ arising from the Calabi ansatz. This suggests geometric hypotheses (which we call ``$\sigma$-constancy'') to impose upon the base metric $g_M$ and Hermitian structure $h$ in order that the scalar curvature of $g$ be specified by solving an ODE. We show that $\sigma$-constancy is ``necessary and sufficient for the Calabi ansatz to work'' in the following sense. Under the assumption of $\sigma$-constancy, the disk bundle admits a one-parameter family of complete Kähler metrics of constant scalar curvature that restrict to $g_M$ on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of $\sigma$-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E).

Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kähler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kähler metric on  $\mathbf{C}^2$.