Lemma 18.39.1EF14C0C90C6C

Proof. For $d=0$, the manifold $M$ is a finite set and the dimension is at most its number of points. Let $d\geq 1$. Fix smooth Hermitian metrics on $E$ and $F$, a smooth positive volume form $\mathrm{d}V$, and finitely many coordinate charts with holomorphic frames, together with smaller subcharts that still cover $M$. For $|j|\leq C_0m$, the local weight of the metric on $E^ m\otimes F^ j$ has second derivatives bounded by $Cm$. At a point $p$ of a smaller subchart, multiply the frame by the exponential of a holomorphic affine function to remove the constant and real-linear terms of this weight at $p$. The new weight vanishes at $p$ and is bounded by $Cm|z-p|^2$ on a fixed neighborhood. A ball of radius $\varepsilon m^{-1/2}$ about $p$ lies in the chart, with $\varepsilon >0$ independent of $p$, $m$ and $j$, and on this ball the weight is bounded. The submean inequality for the holomorphic coefficient of a section $s$ on this ball gives

where all constants depend only on the fixed charts, metrics, volume and $C_0$. Evaluation at $p$ has a one-dimensional target, so for an $L^2$-orthonormal basis $s_1,\ldots ,s_ N$ of $H^0(M,E^ m\otimes F^ j)$ its squared norm is $\sum _{\ell =1}^ N|s_\ell (p)|^2\leq Cm^ d$. Integrating over $M$ gives $N\leq C\operatorname {Vol}(M)m^ d$. $\square$