2004/03/31 by Nobuhiro Honda, Honda, Nobuhiro · 1 citation
Mathematics · Physics and Astronomy · #14C05 #53C25 #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.DG #msc:14C05 #msc:53C25
paper · pdf · doi:10.48550/arxiv.math/0403528
71 pages. V2; errors corrected. V3; 15 figures added
openalex publication_date 2004/03/31 · arxiv created 2006/04/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider self-dual metrics on 3CP2 of positive scalar curvature admitting a non-trivial Killing field, but which is not conformally isometric to LeBrun's metrics. Firstly, we determine defining equations of the twistor spaces of such self-dual metrics. Next we prove that conversely, the complex threefolds defined by the equations always become twistor spaces of self-dual metrics on 3CP2 of the above kind. As a corollary, we determine a global structure of the moduli spaces of these self-dual metrics; namely we show that the moduli space is non-empty and isomorphic to R3/G, where G is an involution of R3 having one-dimensional fixed locus. Combined with works of LeBrun, this settles a moduli problem of self-dual metrics on 3CP2 of positive scalar curvature admitting a non-trivial Killing field. In our proof, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.