Multiplicites and modifications, and singularities associated to blowing down negative vector bundles

We first present the mixed Hilbert-Samuel multiplicities of analytic local rings over $\mathbb{C}$ as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give estimates or an exact formula for the multiplicities of isolated singularities that given by the Grauert blow-downs of negative holomorphic vector bundles.