ScalingStacks

Proof. [046P]

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

Proof.

The new Kähler form ω(2)\omega^{(2)} is cohomologous to ω(1)\omega^{(1)} by the formula

ω−=ω(1)+d⁡(−1​∂φ−∂ηp​d​ηp−−1​∂φ−∂η¯p​d​η¯p).\omega_{-}=\omega^{(1)}+d(\sqrt{-1}\frac{\partial\varphi^{-}}{\partial\eta_{p}}d\eta_{p}-\sqrt{-1}\frac{\partial\varphi^{-}}{\partial\bar{\eta}_{p}}d\bar{\eta}_{p}).

The claim then follows from Lemma 4.14. ∎

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