Overcrowding and hole probabilities for random zeros on complex manifolds Bernard ShiffmanSteve ZelditchScott Zrebiec 32L1032A6032L0560D05positive line bundleBergman kernelSzego kernelrandom zeroslarge deviations 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 random holomorphic section of the $N$-th power of a positive line bundle on a compact K\"ahler 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 $\exp (-C_{U} N^{m + 1})$. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3700 10.1512/iumj.2008.57.3700 en Indiana Univ. Math. J. 57 (2008) 1977 - 1998 state-of-the-art mathematics http://iumj.org/access/