Local Classification of Conformally-Einstein Kähler Metrics in Higher Dimensions

Abstract

m > 3, or (0.2) with m 2, if and only if Q is a speci®c type of a rational function of t.A global classi®cation of all M, g, m, t with (0.1) and m > 3, or (0.2) and m 2, for which M is compact, can similarly be derived from a global classi®cation of compact Ka Èhler manifolds with special Ka Èhler±Ricci potentials. These classi®cation theorems both require extensive additional arguments, based on two different methods, and will therefore appear in separate papers [12,13].The simplest examples of quadruples M; g; m; t with (0.1) for which M is compact, m > 2, and t 0 somewhere in M, are provided by some Riemannian products that have apparently been known for decades; see § 25 below. Another family of compact examples with (0.1), representing all dimensions m > 2, and this time having t T 0 everywhere in M (so that g is globally conformally Einstein), was constructed by Lionel Be Ârard Bergery [2]; cf. also § 26 below. More recent extensions of the results of [2] can be found in [28].Compact Ka Èhler surfaces M; g with (0.1) have been studied by many authors. For instance, Page [21] found the only known example of an Einstein metric on a compact complex surface (namely, on M CP 2 # CP 2 ) which is globally conformally-Ka Èhler, but not Ka Èhler; its conformally-related Ka Èhler metric was independently discovered by Calabi [6,7,9]. Page's manifold thus satis®es (0.1) (and, in fact, (0.2)) with m 2, and has t T 0 everywhere in M. Examples of (0.2) with m 2 such that M is compact and t vanishes somewhere are constructed, for minimal ruled surfaces M, in [16] and [27]. Other results concerning (0.1) for compact Ka Èhler surfaces M; g include LeBrun's structure theorem [20] for Hermitian Einstein metrics on compact complex surfaces and the variational characterization of such metrics in [25]. For non-compact examples, see [1,11].Condition (0.1) with m 2 is much less restrictive than for m > 3, as it implies (0.2) in the latter case, but not in the former. This re¯ects the fact that, at points where the scalar curvature s of a Ka Èhler-surface metric g is non-zero, the metric e g g=s 2 already satis®es a consequence of the Einstein condition (namely, vanishing of the divergence of the self-dual Weyl tensor), and is, up to a constant factor, the only metric conformal to g with this property; see [10, top of p. 417].We wish to thank the referee for suggesting changes that make the present paper easier to read and bringing Gudmundsson's paper [15] to our attention.

Special Kähler–Ricci potentials and Ricci solitons

“…metrics. Special cases of these were given in [6]. They are derived below from the Ricci-Hessian equation (2.4), i.e.…”

“…According to [6,Lemma 11.1a], Q, ∆τ , φ, ψ and µ are locally C ∞ functions of τ on M τ . If g is a standard s.k.r.p.…”

“…The involution above is defined in part via a conformal change, and one can ask whether, starting with a Kähler metric g, one can find a metric g/τ 2 , for some function τ as above, which is a Ricci soliton. 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].…”

“…First, τ is a special Kähler-Ricci potential. This notion (Definition 3.4) was first defined in [6], and implies that τ is a Killing potential, and that (1.1) holds in some (generically nonempty) open set of the manifold. Second, the Kähler-Ricci soliton ( g, τ ) is locally of a type first described by Koiso in [10] (see also [4]).…”

“…After recalling the geometric structure of a Kähler metric admitting a special Kähler-Ricci potential, and presenting a duality result in this context in §5, we summarize our results in §6 by proving Theorem 6.1, from which Theorem A easily follows. Our conventions throughout closely follow [6].…”