Existence of Conical Higher cscK Metrics on a Minimal Ruled Surface
Rajas Sandeep Sompurkar
TL;DR
The paper proves that conical higher cscK metrics exist in every Kähler class on the pseudo-Hirzebruch surface X, constructed via the momentum (Calabi) ansatz with conical singularities along S0 and S∞. It reduces the PDE to a nonlinear ODE for a momentum profile, analyzes boundary-value problems to establish existence and uniqueness of solutions corresponding to appropriate cone angles, and shows these metrics satisfy polyhomogeneous (or conormal) asymptotics. Global curvature is encoded as a top Chern current, enabling a cohomological interpretation and a linear relation between cone angles derived from the top log Bando-Futaki invariant. The results extend the domain of canonical Kähler metrics with conical singularities, providing a concrete, symmetry-based construction on a nontrivial ruled surface and clarifying obstructions via Bedford–Taylor currents and Futaki-type invariants.
Abstract
A higher extremal Kähler metric is defined (motivated by analogy with the definition of an extremal Kähler metric) as one whose top Chern form equals a smooth function multiplied by its volume form such that the gradient of the function is a holomorphic vector field. A special case of this is a higher cscK metric which is defined (again by analogy with the definition of a cscK metric) as one whose top Chern form is a constant multiple of its volume form or equivalently whose top Chern form is harmonic. In our previous paper on higher extremal Kähler metrics we had looked at a certain class of minimal ruled surfaces called as pseudo-Hirzebruch surfaces all of which contain two special divisors (viz. the zero and infinity divisors) and serve as example manifolds in the momentum construction which is used for producing explicit examples of the above-mentioned kinds of canonical metrics. We had proven that every Kähler class on such a surface admits a higher extremal Kähler metric which is not higher cscK and we had further proven by using the top Bando-Futaki invariant that higher cscK metrics do not exist in any Kähler class on the surface. In this paper we will see that if we allow our metrics to develop conical singularities along at least one of the two special divisors then we do get conical higher cscK metrics in each Kähler class by the momentum construction. We will show that our constructed metrics satisfy the polyhomogeneous condition for conical Kähler metrics and we will interpret the conical higher cscK equation globally on the surface in terms of the currents of integration along the two divisors. We will then introduce and employ the top $\log$ Bando-Futaki invariant to obtain a linear relationship between the cone angles of the conical singularities along the two divisors.