Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914692593614848 |
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| author | Sompurkar, Rajas Sandeep |
| author_facet | Sompurkar, Rajas Sandeep |
| contents | In this paper we prove that on a special type of minimal ruled surface, which is an example of a `pseudo-Hirzebruch surface', every Kähler class admits a certain kind of `higher extremal Kähler metric', which is a Kähler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kähler metric. From an already proven result, it will follow that this specific higher extremal Kähler metric cannot be a `higher constant scalar curvature Kähler (hcscK) metric', which is defined, again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric, to be a Kähler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any Kähler class on this surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_07127 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface Sompurkar, Rajas Sandeep Differential Geometry Complex Variables 2020: 53C55 (Primary), 53C25, 32Q15, 32J27, 32J15, 34B30 (Secondary) In this paper we prove that on a special type of minimal ruled surface, which is an example of a `pseudo-Hirzebruch surface', every Kähler class admits a certain kind of `higher extremal Kähler metric', which is a Kähler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kähler metric. From an already proven result, it will follow that this specific higher extremal Kähler metric cannot be a `higher constant scalar curvature Kähler (hcscK) metric', which is defined, again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric, to be a Kähler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any Kähler class on this surface. |
| title | Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface |
| topic | Differential Geometry Complex Variables 2020: 53C55 (Primary), 53C25, 32Q15, 32J27, 32J15, 34B30 (Secondary) |
| url | https://arxiv.org/abs/2302.07127 |