Canonical Kähler metrics on classes of Lorentzian 4-manifolds

Abstract

Conditions for the existence of Kähler–Einstein metrics and central Kähler metrics (Maschler in Trans Am Math Soc 355:2161–2182, 2003) along with examples, both old and new, are given on classes of Lorentzian 4-manifolds with two distinguished vector fields. The results utilize the general construction (Aazami and Maschler in Kähler metrics via Lorentzian geometry in dimension four, Complex Manifolds 7:36–61 (2020) of Kähler metrics on such manifolds. The examples include both complete and incomplete metrics, and some reside on Lie groups associated with four types of Lie algebras. An appendix includes a similar construction for scalar-flat Kähler metrics.

Appendix A. Scalar-flat Kähler surfaces arising from pp-waves

We address here the question of whether it is possible to generate scalar-flat Kähler surfaces with symmetry from an ansatz similar to the one used for admissible metrics and give examples where the background Lorentzian metric is a pp-wave.

Recall LeBrun’s general ansatz [15] for such Kähler metrics,

$$\begin{aligned} g_{\scriptscriptstyle K}=e^uw(dp^2+dq^2)+wdz^2+w^{-1}\theta ^2, \end{aligned}$$

where p, q, z form a coordinate system for a region in $\mathbb {R}^3$, and the manifold M is the total space of a circle bundle over such region (provided the de Rham class of a certain curvature 2-form associated with the connection 1-form $\theta$ is integral). Note that z is also a hamiltonian for a holomorphic Killing field, and the following PDEs hold for u, $w>0$:

$$\begin{aligned} \begin{aligned}&u_{pp}+u_{qq}+(e^u)_{zz}=0\\&w_{pp}+w_{qq}+(we^u)_{zz}=0. \end{aligned} \end{aligned}$$

(46)

With this ansatz in mind, let $\{p,q,z,t\}$ be local coordinates on a manifold M, and fix two smooth functions u(p, q, z), $w(p,q,z)>0$. Consider the 2-form

$$\begin{aligned} \omega (a,b):=d(e^\tau {{\varvec{p}}}^\flat )(a,b) \end{aligned}$$

(47)

where $\tau$ is a smooth function, ${{\varvec{p}}}$ is a vector field with 1-form ${{\varvec{p}}}^\flat$ dual to it with respect to some given semi-Riemannian metric. Let a, b take values in frame fields ${{\varvec{k}}}$, ${{\varvec{t}}}$, ${\varvec{x}}=\partial _p$, ${\varvec{y}}=\partial _q$ residing in the coordinate neighborhood. Mimicking LeBrun’s ansatz, we require $d(e^\tau {{\varvec{p}}}^\flat )(a,b)$ to have the value $e^uw$ when $a={\varvec{x}}$, $b={\varvec{y}}$, the value w when $a={{\varvec{k}}}$, $b={{\varvec{t}}}$, and the value zero for all other pairs a, b with $a\in \{{{\varvec{k}}},{{\varvec{t}}}\}$, $b\in \{{\varvec{x}},{\varvec{y}}\}$.

Furthermore, we define an almost complex structure by $J{{\varvec{k}}}={{\varvec{t}}}$, $J{\varvec{x}}={\varvec{y}}$. Note that the above values on the frame imply that $\omega$ is J-invariant and symplectic. We further require

• J is integrable;

• ${{\varvec{k}}}$ preserves J;

• ${{\varvec{k}}}$ is a hamiltonian vector field for $\omega$, with hamiltonian z;

• The following PDEs hold for u, w:

  $$\begin{aligned}&d^2_{\varvec{x}}u +d^2_{\varvec{y}}u+(e^u)_{zz}=0,\nonumber \\&d^2_{\varvec{x}}w +d^2_{\varvec{y}}w+(we^u)_{zz}=0. \end{aligned}$$

  (48)

If these conditions hold, it will follow from [15] that $g_{\scriptscriptstyle K}=\omega (\cdot , J\cdot )$ is a scalar-flat Kähler metric defined in the coordinate domain.

We now demonstrate by verifying these conditions that examples of this construction hold, in which the semi-Riemannian background metric is a pp-wave, given by

$$\begin{aligned} g_L=H(x,y,u)du^2+2du\odot dv+dx^2+dy^2. \end{aligned}$$

(49)

We choose our frame as follows: ${\varvec{x}}=\partial _x+\partial _y+\partial _v$, ${\varvec{y}}=-\partial _x/2-\partial _y/2+\partial _u-H\partial _v/2$, ${{\varvec{k}}}=\partial _v$, ${{\varvec{t}}}=-3\partial _x-\partial _y-2\partial _v$. Next we take ${{\varvec{p}}}=\partial _x-\partial _u+(H+1)\partial _v/2$, so that

$$\begin{aligned} {{\varvec{p}}}^\flat =\mathrm{d}x+\frac{1-H}{2}\mathrm{d}u-\mathrm{d}v. \end{aligned}$$

Just as in Remark 2.2, integrability of J is checked by verifying $N({{\varvec{k}}},{\varvec{x}})=0$, where N is the Nijenhuis tensor. This holds if and only if $3H_x+H_y=0$. We choose either $H(x,y,u)=e^{x-3y}$ or $H(x,y,u)=x-3y$. In the latter case $g_L$ is a flat pp-wave and the frame Lie brackets satisfy the Lie algebra relations of $\mathfrak {nil}_3\times \mathbb {R}$. We check the remaining conditions only for the latter case, as the former case is similar.

One calculates that for $\tau =\tau (x,y)$, to obtain the above values of $\omega$ on our frame, and hence its J-invariance, $\tau _x+\tau _y=0$ must be required (specifically to have, say, $\omega ({{\varvec{k}}},{\varvec{x}})=0$), and we specialize to the case $\tau =\log \,(y-x)$.

The nonzero values of the of the 2-form on our frame fields are

$$\begin{aligned} \omega ({\varvec{x}},{\varvec{y}})=e^\tau ,\qquad \omega ({{\varvec{k}}},{{\varvec{t}}})=2, \end{aligned}$$

so that we take $w=2$ and $u=\tau -\log 2=\log [(y-x)/2]$ (for $H=e^{x-3y}$ one has instead $\omega ({\varvec{x}},{\varvec{y}})=e^\tau e^{x-3y})$. Also, $\iota _{{\varvec{k}}}\omega =\iota _{\partial _v}\omega =dy-dx$ and thus the hamiltonian associated with ${{\varvec{k}}}$ is $z=y-x$. Now $e^u=(y-x)/2=z/2$, so $(e^u)_{zz}=0$, while $(d^2_{\varvec{x}}+d^2_{\varvec{y}})u=-1/(y-x)^2+1/(y-x)^2=0$. Thus, equations (48) clearly hold, and $g_{\scriptscriptstyle K}$ is scalar-flat Kähler on the region in $\mathbb {R}^4$ given by $\{y>x\}$. In fact, for both choices of H(x, y, u), the metric $g_{\scriptscriptstyle K}$ is also conformally Einstein: $e^{-2\tau }g_{\scriptscriptstyle K}$ is an Einstein metric with scalar curvature $-12$.