We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge-Amp`ere operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and applies to test functions (linear statistics) which are merely Lipschitz continuous. Moreover, a scaling limit of the correlation functions in the bulk is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This geometric setting applies in particular to normal random matrix ensembles, the two dimensional Coulomb gas, free fermions in a strong magnetic field and multivariate orthogonal polynomials.
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