Existence of conical higher cscK metrics on a minimal ruled surface

Abstract

A ‘higher extremal Kähler metric’ is defined (motivated by analogy with the definition of an extremal Kähler metric) as a Kähler metric whose top Chern form equals a globally defined smooth function multiplied by its volume form such that the gradient of the smooth function is a holomorphic vector field. A special case of this kind of metric is a ‘higher constant scalar curvature Kähler (higher cscK) metric’ which is defined (again by analogy with the definition of a constant scalar curvature Kähler (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 the primary example manifolds in the momentum construction method (the Calabi ansatz procedure) which is used for producing explicit examples of the above-mentioned kinds of canonical Kähler metrics. We had proven that every Kähler class on such a surface admits a momentum-constructed 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 of the surface then we do get ‘conical higher cscK metrics’ in each Kähler class of the surface by the momentum construction method. 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 zero and infinity divisors. We will introduce the ‘top Bando-Futaki invariant’ and then try to prove some standard expected results about it, with the final aim being to employ it to arrive at a certain linear relationship, that we conjecture must exist, between the cone angles of the conical singularities along the zero and infinity divisors.