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arXiv · 1509.04585

Classification of polytope metrics and complete scalar-flat K\"ahler 4-Manifolds with two symmetries

Brian Weber
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Abstract

We study unbounded 2-dimensional metric polytopes such as those arising as K\"ahler quotients of complete K\"ahler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible metrics on on the corresponding K\"ahler 4-manifolds. If the polytope is the plane or half-plane then only flat metrics are possible, and if the polytope has one corner then the 2-parameter family of generalized Taub-NUTs (discovered by Donaldson) are indeed the only possible metrics. Polytopes with $n\ge3$ edges admit an $(n+2)$-dimensional family of possible metrics.

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Brian Weber. 2015-09-15. Classification of polytope metrics and complete scalar-flat K\"ahler 4-Manifolds with two symmetries. https://arxiv.org/abs/1509.04585

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