Special Kähler-Ricci potentials on compact Kähler manifolds

Abstract

Abstract. A special Kähler-Ricci potential on a Kähler manifold is any nonconstant C ∞ function τ ι such that J(∇τ ι) is a Killing vector field and, at every point with dτ ι = 0, all nonzero tangent vectors orthogonal to ∇τ ι and J(∇τ ι) are eigenvectors of both ∇dτ ι and the Ricci tensor. For instance, this is always the case if τ ι is a nonconstant C ∞ function on a Kähler manifold (M, g) of complex dimension m > 2 and the metricg = g/τ ι 2 , defined wherever τ ι = 0, is Einstein. (When such τ ι exists, (M, g) may be called almost-everywhere conformally Einstein.) We provide a complete classification of compact Kähler manifolds with special Kähler-Ricci potentials and use it to prove a structure theorem for compact Kähler manifolds of any complex dimension m > 2 which are almost-everywhere conformally Einstein. §0. Introduction This paper, although self-contained, can also be viewed as the second in a series of three papers that starts with [8] and ends with [9].We call τ ι a special Kähler-Ricci potential on a Kähler manifold (M, g) if τ ι is a nonconstant Killing potential on (M, g) and, at every point (0.1) with dτ ι = 0, all nonzero tangent vectors orthogonal to v = ∇τ ι and to u = Jv are eigenvectors of both ∇dτ ι and the Ricci tensor r.(Cf.[8], §7; for more on Killing potentials, see §4 below.) The word 'potential' reflects the fact that (0.1) is closely related, although not equivalent, to the requirement that ∇dτ ι + χ r = σg for some C ∞ functions χ, σ (see [8], beginning of §7). This requirement is reminiscent of Kähler-Ricci solitons (some of which, in fact, do satisfy (0.1), cf.[10] and Remark 10.1 below); while, in complex dimensions m > 2, it implies that τ ι arises from a Hamiltonian 2-form on the underlying Kähler manifold ([2], §1.4). See also [5]. What further sparked our interest in (0.1) was its being, in cases such as (0.4) below, a consequence of the following assumption:(M, g) is a Kähler manifold of complex dimension m and τ ι is (0.2) a nonconstant C ∞ function on M such that the conformally related metricg = g/τ ι 2 , defined wherever τ ι = 0, is Einstein.When m > 2, (0.2) implies the seemingly stronger condition (0.3) M, g, m, τ ι satisfy (0.2) and dτ ι ∧ d∆τ ι = 0 everywhere in M 1991 Mathematics Subject Classification. Primary 53C55, 53C21; Secondary 53C25.

The eta invariant in the doubly Kählerian conformally compact Einstein case

“…We give here an outline of the analog, for manifolds with boundary, of the main theorem in [5] (see Theorem 2 below).…”

“…(4) This requirement on τ is a technical notion which holds in the conformally Einstein case if either the complex dimension of M is at least three, or else it is two and dτ ∧ d g τ = 0, with g the g-Laplacian. Metrics satisfying (4) have been classified both locally and on compact manifolds in [4] and [5]. Specifically, a special Kähler-Ricci potential is a Killing potential on a Kähler manifold, i.e.…”

“…By [5,Lemma 7.3], N is a complex submanifold of M which is of complex codimension one, or else consists of a single point. Furthermore, any critical manifold is the τ -preimage of an extreme value for τ (see the proof of Proposition 11.4 in [5]). As τ is a defining function, the zero level set of τ does not contain any critical points (and is the boundary), so that N is the unique critical manifold of τ .…”

“…These formulas are obtained via moment map techniques, specifically the Duistermaat-Heckman Theorem. The assumptions are either that the Kähler metric admits a special Kähler-Ricci potential in the sense of [4,5], or that it is part of an ambitoric structure, in the sense defined recently by Apostolov et al [1], as well as a toric one. Common to both cases is the existence of a second, oppositely oriented complex structure, with respect to which one has a second Kähler metric in the conformal class, which plays a major role in the derivation of the formulas.…”

Special Kähler–Ricci potentials and Ricci solitons

“…This structure, defined on M τ , extends uniquely to M (see [7,Remark 16.4]), and the corresponding extension of g (see end of §5.1) to M τ −1 (0) is still Kähler with respect to it.…”