Regularization of closed positive currents and extension theory

Abstract

In this note we extend to any bidimension (p,p) the Demailly theorem of regularization of closed positive (1,1)-currents on a compact Kähler manifold X of dimension n. When the manifold X is projective, we get explicitly a closed regularization with bounded negative part, constructed by using the space Cp(X) of effective algebraic cycles of X of dimension p. This space can be injected in the space of divisors of Cn−p−1(X) and we arrive at an intrinsic construction of the Skoda potential associated with a closed positive current of X. On another hand, in the case of a divisor D of X, we give an explicit bound for the degree of an irreducible component of the singular locus Dsing, involving the geometry of X. Lastly when X is embedded in the projective space PN, we prove the existence of a closed current extending in the generalized sense a given closed current of X, by using here a space of cycles. As an application, we obtain a characterization of the cohomology classes contained in some algebraic hypersurface of X.