Bergman kernel asymptotics for big and semi-positive line bundles

Let $L$ be a holomorphic line bundle over a compact complex manifold $M$ and let $L^k$ be the $k$-th power of $L$. If $L$ is semi-positive and positive at some point, we show that the Bergman kernel of $L^k$ admits a full asymptotic expansion on the set where $L$ is positive, with the possible exception of a proper analytic variety of $M$. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric. This yields another proof of the Shiffman conjecture.