Existence of Conical Higher cscK Metrics on a Minimal Ruled Surface

TL;DR

This paper demonstrates the existence of conical higher cscK metrics on minimal ruled surfaces by allowing conical singularities along divisors, extending previous results on smooth metrics.

Contribution

It introduces conical higher cscK metrics on pseudo-Hirzebruch surfaces and establishes a conjectural relationship between cone angles using a new top log Bando-Futaki invariant.

Findings

Conical higher cscK metrics exist in each K"ahler class with singularities along divisors.

Constructed metrics satisfy the polyhomogeneous condition for conical K"ahler metrics.

A conjectural linear relation between cone angles is proposed based on the top log Bando-Futaki invariant.

Abstract

A higher extremal K\"ahler metric is defined (motivated by analogy with the definition of an extremal K\"ahler 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 K\"ahler 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 K\"ahler class on such a surface admits a higher extremal K\"ahler 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 K\"ahler 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 K\"ahler class by the momentum construction. We will show that our constructed metrics satisfy the polyhomogeneous condition for conical K\"ahler 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 introduce the top $\log$ Bando-Futaki invariant and then employ it to arrive at a certain conjectural linear relationship between the cone angles of the conical singularities along the two divisors.