Speaker: Rajas Sandeep Sompurkar, IISc Bengaluru
Venue: Online, https://meet.google.com/mvz-brfo-pmw
Date: Friday , 22nd November, 2024, Time: 02:30 p.m.
Title: The Momentum Construction Method for Higher Extremal Kähler and Conical Higher cscK
Metrics
Abstract: This talk consists of two parts. In both the parts we study two new notions of canonical
Kähler metrics introduced by Pingali viz. `higher extremal Kähler metric' and `higher constant
scalar curvature Kähler (higher cscK) metric' both of whose definitions are analogous to the
definitions of extremal Kähler metric and constant scalar curvature Kähler (cscK) metric
respectively. On a compact Kähler manifold a higher extremal Kähler metric is one whose top
Chern form equals its volume form multiplied by a smooth function whose gradient is a
holomorphic vector field, while a higher cscK metric is one 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 surfaces called as `pseudo-Hirzebruch surfaces' which
are the projective completions of holomorphic line bundles of non-zero degrees over 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 (or the Calabi ansatz procedure) for finding explicit
examples of various kinds of canonical metrics on them.
In the first part of this talk we will see by using the momentum construction method that on a
pseudo-Hirzebruch surface every Kähler class admits a higher extremal Kähler metric which is not
higher cscK. The construction of the required metric boils down to solving an ODE depending on a
parameter on an interval with some boundary conditions, but the ODE is not directly integrable
and requires a very delicate analysis 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 we will finally conclude from this that higher cscK metrics (momentum-constructed or
otherwise) do not exist in any Kähler class on this Kähler surface.
In the second part of this talk we will see that if we allow our metrics to develop `conical
singularities' along at least one of the zero and infinity divisors of a pseudo-Hirzebruch surface
then we do get `conical higher cscK metrics' in each Kähler class of the Kähler surface by the
momentum construction method. 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. We can then see that our momentum-constructed metrics are
conical Kähler metrics satisfying the `polyhomogeneous condition' of Jeffres-Mazzeo-Rubinstein,
and we will be able to interpret the conical higher cscK equation globally on the surface in terms of
the currents of integration along its zero and infinity divisors by using Bedford-Taylor theory.