On the approximation of positive closed currents on compact Kähler manifolds

Let L be a holomorphic line bundle over a compact Kähler manifold X endowed with a singular Hermitian metric h with curvature current c₁ (L, h) ≥ 0. In certain cases when the wedge product c₁ (L,h) ^κ is a well defined current for some positive integer κ ≤ dim X, we prove that c₁ (L,h)^κ can be approximated by averages of currents of integration over the common zero sets of κ-tuples of holomorphic sections over X of the high powers L ^p:= L^⊗p. In the second part of the paper, we study the convergence of the Fubini-Study currents and the equidistribution of zeros of L²-holomorphic sections of the adjoint bundles L^p ⊗ Kx, where L is a holomorphic line bundle over a complex manifold X endowed with a singular Hermitian metric h with positive curvature current. As an application, we obtain an approximation theorem for the current c₁ (L, h)^κ using currents of integration over the common zero sets of κ-tuples of sections of L^p ⊗ Kx.