  Let $X$ be a compact Riemann surface and $\mathcal L$ be a positive line
bundle on it. We study the conditional zero expectation of all the holomorphic
sections of $\mathcal L^n$ which do not vanish on $D$ for some fixed open
subset $D$ of $X$. We prove that as $n$ tends to infinity, the zeros of these
sections are equidistributed outside $D$ with respect to a probability measure
$\nu$. This gives rise to a surprising forbidden set.
