Existence of Conical Higher cscK Metrics on a Minimal Ruled Surface

A `higher extremal K\"ahler metric' is defined (motivated by analogy with the definition of an extremal K\"ahler metric) as a K\"ahler metric whose top Chern form and volume form ``differ'' by a smooth function whose gradient is a holomorphic vector field. A special case of this kind of metric is a `higher constant scalar curvature K\"ahler (higher cscK) metric' which is defined (again by analogy with the definition of a constant scalar curvature K\"ahler (cscK) metric) as a K\"ahler metric whose top Chern form is a real 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 special class of minimal ruled surfaces called as `pseudo-Hirzebruch surfaces' all of which contain two special divisors (viz. the zero and infinity divisors) and have got some nice symmetries which enable the use of the momentum construction method for finding explicit examples of these 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 that higher cscK metrics do not exist in any K\"ahler class on such a surface. In this paper we will see that if we allow our metrics to develop `conical singularities' along at least one of the two divisors then we do get `conical higher cscK metrics' in each K\"ahler class of the surface by the momentum construction method. We will show that our constructed metrics are ``genuine'' conical K\"ahler metrics and we will interpret the conical higher cscK equation globally on the surface in terms of currents. We will then introduce and employ the `top Bando-Futaki invariant' to obtain the linear relationship between the cone angles of the conical singularities of our constructed metrics along the zero and infinity divisors.