Kähler–Einstein Manifolds

Abstract

A Kähler manifold (M, g) is Einstein when there exists $\lambda \in \mathbb {R}$ such that ρ = λω, where ω is the Kähler form associated to g and ρ is its Ricci form. The constant λ is called the Einstein constant and it turns out that λ = s∕2n, where s is the scalar curvature of the metric g and n the complex dimension of M (as a general reference for this chapter see e.g. Tian (Canonical Metrics in Kähler Geometry. Notes taken by Meike Akveld. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2000)). If $\omega =\frac {i}{2} \sum _{j=1}^{n}g_{\alpha \bar {\beta }} dz_{\alpha }\wedge d\bar {z}_{\bar {\beta }}$ is the local expression of ω on an open set U with local coordinates (z ₁, …, z _n) centered at some point p then the Ricci form is the 2-form on M of type (1, 1) defined by

$$\displaystyle \begin{aligned} \rho =-i\partial\bar{\partial}\log\det g_{\alpha\bar{\beta}}. \end{aligned} $$

(4.1)

By the $\partial \bar {\partial }$-Lemma (and by shrinking U if necessary) this is equivalent to require that

$$\displaystyle \begin{aligned} \det(g_{\alpha\bar{\beta}})=e^{-\frac{\lambda}{2}\mathrm{D}_0(z)+f+\bar f}, \end{aligned} $$

(4.2)

for some holomorphic function f, where D_p denotes Calabi’s diastasis function centered at p. In this chapter we study Kähler immersions of Kähler–Einstein manifolds into complex space forms . We begin describing in the next section the work of Umehara (Tohoku Math J 39:385–389, 1987) which completely classifies Kähler–Einstein manifolds admitting a Kähler immersion into the finite dimensional complex hyperbolic or flat space. In Sect. 4.3 we summarize what is known about Kähler immersions of Kähler–Einstein manifolds into the finite dimensional complex projective space.