ScalingStacks

Theorem 4.38 . [046X]

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

Theorem 4.38.

(Ooguri-Vafa type metric on the negative vertex) Fix k,α,κk,\alpha,\kappa and 0<ϵ≪10<\epsilon\ll 1, and let 1≪ν≪A3/81\ll\nu\ll A^{3/8}. Then there is an S1S^{1}-invariant Calabi-Yau metric on Mν−M^{-}_{\nu} with S1S^{1}-invariant Kähler potential ϕ−\phi^{-},

ω−=ω(2)+−1​∂∂¯​ϕ−,ω−3=34​−1​Ω∧Ω¯,\omega_{-}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{-},\quad\omega_{-}^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega},

with metric deviation estimate ‖∇g(2)2ϕ−‖C−1−ϵα​(ℬν−)≤Cν2A−3/4.\left\lVert\nabla^{2}_{g^{(2)}}\phi^{-}\right\rVert_{C^{\alpha}_{-1-\epsilon}(\mathcal{B}^{-}_{\nu})}\leq C\nu^{2}A^{-3/4}. The constants depend only on α,ϵ,κ\alpha,\epsilon,\kappa and the scale invariant ellipticity bound on ap​q¯a_{p\bar{q}}.

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