Regularization of closed positive currents and extension theory
Résumé
In this note we extend to any bidimension (p, p) the Demailly theorem of regularization of
closed positive (1, 1)-currents on a compact K¨ahler 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
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