ScalingStacks

Lemma 6 [029G]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Lemma 6

. Let (X,ω)(X,\omega) be a compact Kähler manifold of complex dimension nn, let hh be a smooth function such that ∫Xωn=∫Xeh​ωn\int_{X}\omega^{n}=\int_{X}e^{h}\omega^{n} and φ∈𝒫ω\varphi\in{\cal P}_{\omega} a solution of the complex Monge-Ampère equation

(ω+i​∂∂¯​φ)n=eh+λ​φ​ωn,\displaystyle(\omega+i\partial\bar{\partial}\varphi)^{n}=e^{h+\lambda\varphi}\omega^{n}\,, (4.5)

λ>0\lambda>0. Consider also two solutions φ′​φ′′∈𝒫ω\varphi^{\prime}\,\varphi^{\prime\prime}\in{\cal P}_{\omega} of the complex Monge-Ampère equation (ω+i​∂∂¯​φ)n=eh​ωn(\omega+i\partial\bar{\partial}\varphi)^{n}=e^{h}\omega^{n} such that minX⁡φ′=0=maxX⁡φ′′\min_{X}\varphi^{\prime}=0=\max_{X}\varphi^{\prime\prime}. Then φ′′≤φ≤φ′\varphi^{\prime\prime}\leq\varphi\leq\varphi^{\prime}.

Original mathematics by the credited authors. Source-backed reader collection; mathematical self-containment is not assessed.