On conformally Kähler, Einstein manifolds

Abstract

We prove that any compact complex surface with c 1 > 0 c_1>0 admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blow-up C P 2 # 2 C P 2 ¯ \mathbb {CP}_2\# 2\overline {\mathbb {CP}_2} of the complex projective plane at two distinct points.

Sharp version of the Goldberg–Sachs theorem

“…The more complicated terms such as 'multiple principal totally null field of 2-planes having locally constant real index' will be explained in Sect. 3.…”

“…If the metric has Lorentzian signature (+, +, +, −) then we chose the basis so that g(e 1 , e 1 ) = g(e 2 , e 2 ) = g(e 3 , e 3 ) = 1 = −g(e 4 , e 4 ), and as an example of N we take…”

“…In the case of split signature (+, +, −, −), we have g(e 1 , e 1 ) = g(e 2 , e 2 ) = 1, g(e 3 , e 3 ) = g(e 4 , e 4 ) = −1, and we distinguish two different classes of 2-dimensional totally null N s. As an example of the first class, we take…”

Compact anti-self-dual orbifolds with torus actions

“…The proof of Theorem B. Using Theorem A it is a simple task to classify asymptotically locally Euclidean (ALE) scalar-flat Kähler toric 4-orbifolds, following the procedure suggested by Chen, LeBrun and Weber in [5]. Definition 12.3 (Joyce [15]).…”

Special Kähler–Ricci potentials and Ricci solitons

“…The case where g/τ 2 is Einstein, was the subject of the study of [6,7,8], where local and global classifications were given, and, in all even dimensions larger than four, τ turns out to be, in fact, a special Kähler-Ricci potential. In dimension four this need not be the case, and different compact examples where recently shown to exist in [5].…”