The Momentum Construction Method for Higher Extremal Kähler and Conical Higher cscK Metrics

Abstract

This thesis consists of two parts. In both the parts we study two new notions of canonical K\"ahler metrics on compact K\"ahler manifolds which were introduced by Pingali, viz. `higher extremal K\"ahler metric' and `higher constant scalar curvature K\"ahler (higher cscK) metric', both of whose definitions were formulated by drawing analogies to the definitions of extremal K\"ahler metric and constant scalar curvature K\"ahler (cscK) metric respectively. On a compact K\"ahler manifold a higher extremal K\"ahler metric is a K\"ahler metric whose corresponding top Chern form equals its corresponding volume form multiplied by a smooth real-valued function whose gradient is a holomorphic vector field, while a higher cscK metric is 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 both the parts we consider a special family of minimal ruled complex surfaces called as `pseudo-Hirzebruch surfaces' which are the projective completions of holomorphic line bundles of non-zero degrees over compact Riemann surfaces of genera greater than or equal to two. These surfaces have got some nice symmetries in terms of their fibres and their zero and infinity divisors which enable the use of the momentum construction method of Hwang-Singer (which is a refinement of the Calabi ansatz procedure) for finding explicit examples of various kinds of canonical K\"ahler metrics on them. The momentum construction method converts the PDE defining the kind of canonical K\"ahler metric under consideration into an ODE depending upon some parameters on a closed and bounded interval with some boundary conditions. \par In the first part of this thesis we will prove by using the momentum construction method that on a pseudo-Hirzebruch surface every K\"ahler class admits a higher extremal K\"ahler metric which is not a higher cscK metric. As mentioned above the construction of the required metric boils down to solving an ODE BVP, but the ODE is not directly integrable and the BVP is dependent upon two independent parameters, and hence a very delicate analysis is required for getting the existence of a solution satisfying all the boundary conditions. Then by doing a certain set of computations involving the top Bando-Futaki invariant (which provides an algebro-geometric obstruction to the existence of higher cscK metrics) we will finally conclude that higher cscK metrics (momentum-constructed or otherwise) do not exist in any K\"ahler class on this K\"ahler surface. We will briefly compare the results obtained in this problem with the analogous problem of constructing (usual) extremal K\"ahler metrics which are not (usual) cscK metrics on a pseudo-Hirzebruch surface which has been previously studied by T{\o}nnesen-Friedman and Apostolov-Calderbank-Gauduchon-T{\o}nnesen-Friedman. \par Since we proved in the first part of this thesis that smooth higher cscK metrics do not exist on a pseudo-Hirzebruch surface, in the second part of this thesis we will attempt to construct higher cscK metrics with `conical singularities' along the zero and infinity divisors of the minimal ruled complex surface by the momentum construction method. We will see that in every K\"ahler class of the K\"ahler surface momentum-constructed `conical higher cscK metrics' exist with some values of the cone angles along the zero and infinity divisors of the surface. Even in this case the construction of the required metric boils down to solving a very similar ODE on the same interval but with different parameters and slightly different boundary conditions, and hence having a very similar analysis as in the first part. We will show that our momentum-constructed conical K\"ahler metrics are conical K\"ahler metrics satisfying the strongest condition for conical K\"ahler metrics, viz. the `polyhomogeneous condition' of Jeffres-Mazzeo-Rubinstein in which the local coordinate expression for the metric has a certain special kind of asymptotic power series expansion along the divisors of the conical singularities. We will also interpret the conical higher cscK equation globally on the pseudo-Hirzebruch surface in terms of currents of integration along the zero and infinity divisors of the surface by using Bedford-Taylor theory. We will see that this global interpretation of the conical higher cscK equation resembles the equation of currents giving the global interpretation of the conical (usual) cscK equation obtained by Hashimoto for momentum-constructed conical (usual) cscK metrics on a(n) (actual) Hirzebruch surface.