Einstein Metrics via Derdziński Duality

Abstract Drawing on results of Derdziński’s from the 1980s, we classify conformally Kähler, U (2)-invariant, Einstein metrics on the total space of $${{\mathcal {O}}}(-m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , for all $$m \in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . This yields infinitely many 1-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface $$\mathbb {H}_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> with a cone singularity along the “divisor at infinity”. This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci–flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on $${\mathbb{C}\mathbb{P}}^2 \# \overline{{\mathbb{C}\mathbb{P}}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>#</mml:mo> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric. Such metrics cannot be obtained by applying Derdziński’s theorem.