On the approximation of positive closed currents on compact Kaehler manifolds

Original · EN

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)ᵏ is a well defined current for some positive integer k≤ X, we prove that c₁(L,h)ᵏ can be approximated by averages of currents of integration over the common zero sets of k-tuples of holomorphic sections over X of the high powers Lᵖ:=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 Lᵖ⊗ 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)ᵏ using currents of integration over the common zero sets of k-tuples of sections of Lᵖ⊗ Kₓ.