In this talk we will first see the definitions of ‘higher extremal Kahler metric’ and ‘higher constant scalar curvature Kahler (hcscK) metric’, both of which are analogous to the definitions of extremal Kahler metric and constant scalar curvature Kahler (cscK) metric respectively. Informally speaking, on a compact K ̈ahler manifold a higher extremal Kahler metric is a Kahler metric whose corresponding top Chern form and volume form differ by a smooth real-valued function whose gradient is a holomorphic vector field, and an hcscK metric is a Kahler metric whose top Chern form and volume form differ by a real constant or equivalently whose top Chern form is harmonic. We will then prove that on a special type of minimal ruled complex surface, which is an example of a ‘pseudo-Hirzebruch surface’, every Kahler class admits a higher extremal Kahler metric which is constructed by using the well-known momentum construction method involving the Calabi ansatz procedure. We will then check that this specific higher extremal Kahler metric yielded by the momentum construction method cannot be an hcscK metric. By doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude that hcscK metrics do not exist in any Kahler class on this Kahler surface. We will then see briefly what changes in the calculations in the momentum construction method when we take a general pseudo- Hirzebruch surface which is basically the projectivization of a certain kind of rank two holomorphic vector bundle over a compact Riemann surface of genus greater than or equal to two. It can be seen that the results about the existence of higher extremal Kahler metrics and the non-existence of hcscK metrics obtained in the special case of our minimal ruled surface can be generalized to all pseudo-Hirzebruch surfaces. If time permits we will see the motivation for studying this problem and its analogy with the related and previously well- studied problem of constructing extremal Kahler metrics on a pseudo-Hirzebruch surface.

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Last updated: 09 Sep 2024