Lemma 18.39.1246F892FF98FE

Proof. For $d=0$ the dimension is at most the number of points of $M$. For $d\geq 1$, use fixed smooth metrics, a smooth positive volume form and finitely many framed charts as in the proof of lemma 18.39.1. The local weight of the metric is $m\phi _ E+\sum _ i j_ i\phi _{F_ i}$, whose second derivatives are bounded by $Cm$ uniformly over the stated range. After its constant and real-linear terms at a point have been removed by a holomorphic affine change of frame, it is bounded by a fixed constant on a ball of radius $\varepsilon m^{-1/2}$ about that point. The submean inequality then gives $|s(x)|^2\leq Cm^ d\| s\| _{L^2}^2$, uniformly in $x$, $m$ and the integers $j_ i$. Since evaluation has a one-dimensional target, an orthonormal basis satisfies $\sum _\ell |s_\ell (x)|^2\leq Cm^ d$, and integration gives the estimate. The constants depend only on the fixed bundles, metrics, charts, volume and $C_0$. $\square$