Bergman kernels and equidistribution for sequences of line bundles on K\"ahler manifolds

Given a sequence of positive Hermitian holomorphic line bundles $(L_p,h_p)$ on a K"ahler manifold $X$, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of $L_p$, under a natural convergence assumption on the sequence of curvatures $c_1(L_p,h_p)$. We then apply this to study the asymptotic distribution of common zeros of random sequences of $m$-tuples of sections of $L_p$ as $p\to \mathrm{\infty }$.