ScalingStacks

Theorem 4.2 . [027V]

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 4.2.

We assume that LL is ample and hh is a semipositive continuous metric of LanL^{\mathrm{an}}. Fix a closed subscheme YY, l∈H0​(Y,L|Y)l\in H^{0}(Y,\left.{L}\right|_{{Y}}) and ϵ∈ℝ>0\epsilon\in\mathbb{R}_{>0}. Then there is a positive integer n0{n_{0}} such that, for all n≥n0n\geq{n_{0}}, we can find s∈H0​(X,L⊗n)s\in H^{0}(X,L^{\otimes n}) with

s|Y=l⊗nand‖s‖hn≤en​ϵ​(‖l‖Y,h)n.\left.{s}\right|_{{Y}}=l^{\otimes n}\quad\text{and}\quad\|s\|_{h^{n}}\leq e^{n\epsilon}(\|l\|_{Y,h})^{n}.

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