  We give asymptotic large deviations estimates for the volume inside a domain
U of the zero set of a random polynomial of degree N, or more generally, of a
holomorphic section of the N-th power of a positive line bundle on a compact
Kaehler manifold. In particular, we show that for all $\delta>0$, the
probability that this volume differs by more than $\delta N$ from its average
value is less than $\exp(-C_{\delta,U}N^{m+1})$, for some constant
$C_{\delta,U}>0$. As a consequence, the "hole probability" that a random
section does not vanish in U has an upper bound of the form
$\exp(-C_{U}N^{m+1})$.
