Self-dual metrics and twenty-eight bitangents

Nobuhiro Honda

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

We determine a global structure of the moduli space of self-dual metrics on 3CP2 satisfying the following three properties: (i) the scalar curvature is of positive type, (ii) they admit a non-trivial Killing field, (iii) they are not conformal to the LeBrun’s selfdual metrics based on the ‘hyperbolic ansatz’. We prove that the moduli space of these metrics is isomorphic to an orbifold R3/G, where G is an involution of R3 having two-dimensional fixed locus. In particular, the moduli space is non-empty and connected. We also remark that Joyce’s self-dual metrics with torus symmetry appear as a limit of our self-dual metrics. Our proof of the result is based on the twistor theory. We first determine a defining equation of a projective model of the twistor space of the metric, and then prove that the projective model is always birational to a twistor space, by determining the family of twistor lines. In determining them, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.

Original languageEnglish
Pages (from-to)175-258
Number of pages84
JournalJournal of Differential Geometry
Volume75
Issue number2
DOIs
Publication statusPublished - 2007

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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