Knowlpedia

Definition

Local Kähler potential

A real smooth function whose complex Hessian is a given Kähler form on a coordinate neighborhood.

Definition

Let ω\omega be a on a XX, and let U⊆XU\subseteq X be open. A local Kähler potential for ω\omega on UU is a real-valued smooth function φ:U→R\varphi:U\to\mathbb R satisfying

ω∣U=i∂∂ˉφ.\omega|_U=i\partial\bar\partial\varphi.

Equivalently, using the convention dc=i(∂ˉ−∂)d^c=i(\bar\partial-\partial), one has ω∣U=12ddcφ\omega|_U=\tfrac12dd^c\varphi. The complex Hessian (∂2φ/∂zj∂zˉk)(\partial^2\varphi/\partial z^j\partial\bar z^k) must be positive definite. Every Kähler form admits such potentials on sufficiently small coordinate neighborhoods.

Nonuniqueness

If φ\varphi and ψ\psi are potentials for the same form on UU, then

∂∂ˉ(φ−ψ)=0.\partial\bar\partial(\varphi-\psi)=0.

Thus their difference is pluriharmonic. On a simply connected coordinate neighborhood, it is locally the real part of a holomorphic function. Adding a constant or the real part of a holomorphic function therefore leaves the Kähler form unchanged. Potentials are local scalar descriptions, not canonical global functions.

Examples

On Cn\mathbb C^n,

φ(z)=∑j=1n∣zj∣2\varphi(z)=\sum_{j=1}^n|z^j|^2

is a potential for the standard form i∑jdzj∧dzˉji\sum_j dz^j\wedge d\bar z^j. On the affine chart of , log⁡(1+∣z∣2)\log(1+|z|^2) is a potential for the Fubini–Study form, up to the normalization chosen for that form; Demailly gives the normalized formula in Chapter VI, §4, Example 4.4.

Local versus global

A global Kähler potential is much stronger than local existence. A positive-dimensional compact cannot satisfy ω=i∂∂ˉφ\omega=i\partial\bar\partial\varphi globally, because ω\omega would be exact and would force ∫Xωn=0\int_X\omega^n=0. Local potentials nonetheless glue up to pluriharmonic differences and encode the by

gjkˉ=∂2φ∂zj∂zˉk.g_{j\bar k}=\frac{\partial^2\varphi}{\partial z^j\partial\bar z^k}.
References
  1. Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: §3.1, local potentials for Kähler metrics.
  2. Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter VI, §4, especially Example 4.4 and Theorem 4.8.