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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Distinguished Kähler metrics on Hirzebruch surfaces

Authors: Andrew D. Hwang and Santiago R. Simanca
Journal: Trans. Amer. Math. Soc. 347 (1995), 1013-1021
MSC: Primary 58E11; Secondary 32J27, 53C55
MathSciNet review: 1246528
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Abstract: Let $ {\mathcal{F}_n}$ be a Hirzebruch surface, $ n \geqslant 1$. Using the family of extremal metrics on these surfaces constructed by Calabi [1], we study a closely related scale-invariant variational problem, and show that only $ {\mathcal{F}_1}$ admits an extremal Kähler metric which is critical for this new functional. Applying a result of Derdzinski [3], we prove that this metric cannot be conformally equivalent to an Einstein metric on $ {\mathcal{F}_1}$. When $ n = 2$, we show there is a critical orbifold metric on the space obtained from $ {\mathcal{F}_2}$ by blowing down the negative section.

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Keywords: Complex surface, Hirzebruch space, ruled surface, extremal Kähler metric, Futaki character
Article copyright: © Copyright 1995 American Mathematical Society