Giovanni Bassanelli

A cut-off theorem for plurisubharmonic currents

We show how plurisubharmonic currents can be studied by means of a suitable modification of Federers 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, then the cut-off χγΤ by an analytic subset Υ of Ω is the current of Integration/[F], for a suitable plurisubharmonic function/on Y. 1991 Mathematics Subject Classification: 32F05; 32C30, 58A25. Introduction In this paper we continue the study of plurisubharmonic currents begun in [AB2]. Troughout this note Ω is an open subset of the complex euclidean space C. Let Γ be a current on Ω (i.e. a differential form with distribution coefficients) of bidimension (/?,/?), l < p < N; Tiscalled plurisubharmonicifidc)T> 0. Inparticular if p = N, then Γ is a plurisubharmonic function; indeed a plurisubharmonic function on Ω is nothing but a distribution / such that its Levi form idfif> 0. From this condition, it follows/e L}oc (Ω) and that/coincides, outside a subset of measure zero, with a function which is plurisubharmonic in the classical sense; therefore/is locally essentially bounded from above and its derivatives also belong t o L}oc(i2). All these nice properties do not hold if p < N: if S is a current of bidimension (p, p 4-1) then dS -f dS is in fact pluriharmonic (i.e. f cF-closed) but, in general, it does not have measure coefficients. Therefore, in order to get interesting results, it can be useful to specify the above, so general, definition of plurisubharmonic current adding some other conditions; s it is done, for example, in [Sb] Chapter II, where pluripositive currents are studied. Recall that a current Tis pluripositive if Tis positive or negative, plurisubharmonic This work is partially supported by MURST, 40%.

Lelong numbers of positive plurisubharmonic currents

AbstractA (k,k)-current T on an open subset of CN is plurisubharmonic if is positive. Positive plurisubharmonic currents admit Lelong numbers; we prove here that they are independent on the coordinates system. Moreover, if Y is an analytic subset of pure dimension, then the Lelong numbers of T on Y are given by a non negative weakly plurisubharmonic function.

Transforms of currents by modifications and 1-convex manifolds

Let ′ and be complex manifolds (not compact, a priori), and ′ α → a proper modification with center and exceptional divisor , whose irreducible components are { }. Let be an analytic subset of without irreducible components in : then its strict (proper) transform ′ is a well-defined analytic subset of . In particular, when is a complex hypersurface of , we can define the strict transform ′ and also the total transform

1 -convex manifolds are p- kähler

We study here Kähler-type properties of 1-convex manifolds, using the duality between forms and compactly supported currents, and some properties of the Aeppli groups of (q-convex manifolds. We prove that, when the exceptional setS of the l-convex manifoldX has dimensionk, X is p-Kähler for everyp > k, and isk-Kähler if and only if “the fundamental class” ofS does not vanish. There are classical examples whereX is notk-Kähler even with a smoothS, but we prove that this cannot happen if2k ≥n = dimX, nor for suitable neighborhoods of S; in particular,X is always balanced (i.e.,(n - 1)-Kahler).