2011/03/31 by Andrzej Derdziński, Andrzej Derdzinski
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Materials science #Mathematics #Surface (topology) #math.DG #msc:53C25 #msc:53C55
paper · pdf · doi:10.1007/s00605-011-0345-x
published as Monatshefte für Mathematik 167 (2012), no. 3-4, pp. 431-448 · 16 pages
arxiv created 2011/03/31 · openalex publication_date 2011/10/10 · arxiv updated 2012/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two Kähler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tangent bundle. A special biconformal change is called nontrivial if the two metrics are not each other's constant multiples. For instance, according to a 1995 result of LeBrun, a nontrivial special biconformal change exists for the conformally-Einstein Kähler metric on the two-point blow-up of the complex projective plane, recently discovered by Chen, LeBrun and Weber; the real-holomorphic vector field involved is the gradient of its scalar curvature. The present paper establishes the existence of nontrivial special biconformal changes for some canonical metrics on Del Pezzo surfaces, viz. Kähler-Einstein metrics (when a nontrivial holomorphic vector field exists), non-Einstein Kähler-Ricci solitons, and Kähler metrics admitting nonconstant Killing potentials with geodesic gradients.