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Abstract.Small deformations of Kahler manifolds are Kahler too; we prove here that this is not true for balanced manifolds nor, more generally,

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We show how plurisubharmonic currents can be studied by means of a suitable modification of Federer's theory of flat currents. The goal of the paper is to show that if T is a positive plurisubharmonic current on an open subset Ω of C N , then the cut-off χ γ Τ by an analytic subset Υ of Ω is the current of Integration/[F], for a suitable plurisubharmonic function/on Y.

Some extension problems are considered here for the class of plurisubharmonic currents, i.e. real currents T such that idfiT is positive. In particular we prove the following theorem: "Let Ω be an open subset of C N and Fan analytic subset of Ω. Suppose Tis a negative plurisubharmonic current on Ω -Υ of bidimension (p,p)\ if dim Y