Annotated Definitions

# Kähler manifolds

A Kähler metric on a Riemmanian manifold is one which has especially nice (indeed, bemerkenswert) properties, which are usually summarized by saying that it has compatible Riemannian, symplectic, and complex structures. More precisely (and less transparently), a Kähler structure on a Riemannian manifold $(M^n, g)$ is a pair $(\Omega, J)$ where

• J is a complex structure, i.e. a field of endomorphisms of the tangent bundle TM satisfying $J^2 = -\mathrm{id}$ which is also integrable;
• $\Omega$ is a symplectic 2-form (i.e. $d\Omega = 0$);
• $g(X,Y) = g(JX, JY)$ for all $X, Y \in TM$ (i.e. J makes g a Hermitian metric, or the complex and Riemannian structures play nice with each other), and
• $\Omega(X,Y) = g(JX, Y)$ (a compatibility condition for the symplectic structure).

For a Kähler metric, the Hermitian metric tensor g is specified by a unique function u, in the sense that $g_{\alpha\bar{\beta}} = \frac{\partial^2 u}{\partial z_\alpha \partial \bar{z_\beta}}$. This gives rise to simple explicit expression for the Christoffel symbols and the Ricci and curvature tensors, and “a long list of miracles occur then.” (quoted from Moroianu’s notes, which presumably quote from Kähler’s original paper.)

Kähler metrics  may also be characterized, analytically, by the existence of holomorphic normal coordinates around each point.

Some examples of Kähler manifolds (i.e. manifolds which admit Kähler metrics): $\mathbb{C}^m$ with the usual Hermitian metric; any Riemann surface; complex projective space with the Fubini-Study metric; any projective manifold; more examples from algebraic geometry …

Moroianu’s notes contains much more on Kähler manifolds, including elaborations and proofs of many of the assertions above.