Let $X$ be a compact complex manifold of complex dimension $n\ge 1$, and let $L \to X$ be a positive holomorphic line bundle. For each integer $m\ge 0$, let $L^m$ denote the $m$-fold tensor power of $L$, with $L^0=\mathcal O_X$. For $0\le q\le n$, let $H^q(X,L^m)$ denote the $q$-th sheaf cohomology group of the sheaf of holomorphic sections of $L^m$, set $h^q(X,L^m):=\dim_{\mathbb C}H^q(X,L^m)$, and set $\chi(X,L^m):=\sum_{q=0}^{n}(-1)^q h^q(X,L^m)$. Let $[X]$ denote the fundamental class of $X$, and write $\langle \beta,[X]\rangle$ for evaluation of a degree-$2n$ cohomology class $\beta$ on $[X]$. Then, as $m\to\infty$ through positive integers,

\begin{align*} \chi(X,L^m)=\frac{1}{n!}\left\langle c_1(L)^n,[X]\right\rangle m^n+O(m^{n-1}). \end{align*}

Moreover, for all sufficiently large integers $m$,

\begin{align*} h^0(X,L^m)=\frac{1}{n!}\left\langle c_1(L)^n,[X]\right\rangle m^n+O(m^{n-1}). \end{align*}