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The Monge-Ampère operator and slicing of closed positive currents. (Opérateur de Monge-Ampère et tranchage des courants positifs fermés.) (French) Zbl 1005.32023
Let $$k\leq p\leq n$$ and $$R$$ a current of bidimension $$(p,p)$$ in the polydisk $$\Delta^n= \Delta^k\times \Delta^{n-k}$$ in $$\mathbb{C}^n$$. Let $$\alpha$$ be a measurable, bounded function with compact support in $$\mathbb{C}^k$$, such that $$\int_{\mathbb{C}^k}\alpha= 1$$. The slice of $$R$$ at $$a\in\Delta^k$$, denoted $$\langle R,\pi,a\rangle_\alpha$$, where $$\pi(z)= (z_1,\dots, z_k)= z'$$, is the limit as $$\varepsilon\to 0$$ of $R\wedge \pi^*\Biggl({1\over \varepsilon^{2k}} \alpha\Biggl({z'- a\over\varepsilon}\Biggr)\cdot {1\over 4^k k!} (dd^c|z'|^2)^k\Biggr),$ when this exists. This agrees with the definition of Federer when $$\alpha$$ is the characteristic function of the unit ball in $$\mathbb{C}^k$$, and with Harvey and Shiffman for a smooth compactly supported function. The geometric idea of the slice is that when $$R$$ is the current of integration on an analytic set $$X$$ in $$\Delta^n$$, then outside a countable union of analytic sets in $$\Delta^k$$ the slice is given by the integration on the fiber $$X\cap \pi^{-1}(a)$$ (proved in section 4 of the paper).
Let $$T$$ be a positive, closed current of bidimension $$(p,p)$$ in a neighborhood of the closed unit polydisk $$\overline{\Delta^n}$$. The main result of the paper is that then there exists a pluripolar set $$E$$ in $$\Delta^k$$, independent of $$\alpha$$, such that for all $$a\in \Delta^k\setminus E$$ the slice $$\langle T,\pi, a\rangle_\alpha$$ exists and is in independent of $$\alpha$$. For a smooth $$(p- k,p-k)$$-form $$\varphi$$ in $$\Delta^n$$ and a locally bounded plurisubharmonic function $$v$$ in $$\Delta^k$$ the formula $\int_{\Delta^n} T\wedge (dd^c(v\circ \pi))^k\wedge\varphi= \int_{a\in\Delta^k}\langle T,\pi, a\rangle(\varphi)(dd^c v)^k$ gives the connection between the current $$T$$ and its slice, and generalizes a formula by Federer (where $$v(z)=|z'|^2$$).
The authors also show that for any given pluripolar set $$E$$ in $$\Delta^k$$ there exists a closed, positive current of bidimension $$(p,p)$$ for which the slice does not exist on $$E$$, and that the Lelong number is preserved for the slice, outside a pluripolar exceptional set.
As an application a simplification is given for the proof of the chain theorem from another paper of the same authors [C. R. Acad. Sci., Paris, Sér. I 316, No. 11, 1173-1176 (1993; Zbl 0789.32008)] about the prolongation by zero of a positive, closed current over a closed, complete pluripolar set in $$\Delta^n$$. The conditions are on boundedness of the current in a neighborhood of $$\Delta^k\times \{0\}$$ and boundedness of the slice of $$T$$ on a non-pluripolar set. Moreover, a sufficient condition for the existence of the slice at $$a$$ is shown to be that the Newtonian potential in $$\mathbb{C}^k= \mathbb{R}^{2k}$$ centered at $$a$$ is in $$L^1_{\text{loc}}(\Delta^n,\sigma_T')$$, where $$\sigma_T'= T\wedge(dd^c|z'|^2)^k\wedge (dd^c|z''|^2)^{p- k+1}$$ is the trace measure of $$T\wedge (dd^c|z'|^2)^k$$.
In the proof, the authors use the local potential $$U$$ for the current. For a positive closed current $$T$$ of bidimension $$(p,p)$$ in an open set $$\Omega_1\subset \mathbb{C}^n$$, $$\Omega\subset\subset \Omega_1$$, $$\eta\in{\mathcal D}(\Omega_1)$$, $$0\leq \eta\leq 1$$ and $$\eta\equiv 1$$ in a neighborhood of $$\overline\Omega$$, $$U= U(\Omega, T)= U(\eta, T)$$ is the negative current of bidimension $$(p+ 1,p+1)$$ defined by $U(z)= -{1\over (n-1)(4\pi)^n} \int_{x\in \mathbb{C}^n} \eta(x) T(x)\wedge {(dd^2(|z-x|^2))^{n- 1}\over|z- x|^{2n- 2}}.$ The current $$U$$ has coefficients in $$L^{1+{1\over n}}_{\text{loc}}$$, and they show that $$T- dd^cU\in {\mathcal C}^\infty$$. The main theorem is shown first for regular $$T$$, and with the help of $$U$$ they study the convergence of regularizations of $$T$$. Let the mollifier $$\gamma$$ be a bounded, measurable function with compact support such that $$\int_{\mathbb{C}^n} \gamma=1$$, and $$\gamma_j(|z|)= j^{2n}\gamma(j|z|)$$. For a sequence $$\{v_j\}$$ of plurisubharmonic functions decreasing to a subharmonic function $$v$$ whose unbounded locus locally avoids $$\text{supp }T$$, they show that $(T* \gamma_j)\wedge (dd^c v_j)^k\to T\wedge(dd^c v)^k$ weakly.

##### MSC:
 32U40 Currents 32C30 Integration on analytic sets and spaces, currents 32U25 Lelong numbers 32W20 Complex Monge-Ampère operators
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##### References:
  Alessandrini, L. and Bassanelli, G. Lelong numbers of positive plurisubharmonic currents, preprint. Trento, 1–35, 1991. · Zbl 0735.32009  Blel, M. Potentiel et cône tangent associés à un courant positif fermé dans $$\mathbb{C}$$ n , Thèse d’état, Fac. Sc. Monastir, 1992.  Ben Messaoud, H. Courants intermédiaires associés à un courant positif fermé | Séminaire Lelong-Skoda,Lectures Notes in Math.,1028, 41–68.  Ben Messaoud, H. et EL Mir, H. Tranches et prolongement des courants positifs fermés,CRAS Paris, t316, série I, 1173–1176, 1993.  Ben Messaoud, H. et EL Mir, H. Opérateur de Monge-Ampère et formule de tranchage pour un courant positif fermé,CRAS Paris, 3, t 321, serie I, 1995.  Ben Messaoud, H. et EL Mir, H. Tranchage et prolongement des courants positifs fermés, préprint, Monastir, 1995.  Ben Messaoud, H. et EL Mir, H. Opérateur de Monge-Ampère et potentiels, preprint, Monastir, 1995.  Bedford, E. et Taylor, B.A. A new capacity for plurisubharmonic functions,Acta Math.,149, 1–41, 1982. · Zbl 0547.32012  Blel, M., Demailly, J.P., et Mouzaly, M. Sur l’existence du cône tangent à un courant positif fermé,Ark. for Math.,28(2), 231–248, (1990). · Zbl 0724.32005  Cegrell, U. An estimate of the complex Monge-Ampère operator,Lect. Notes in Math.,1039, 84–87. · Zbl 0529.35018  Chirka, E.M.Complex Analytic Sets, Math. and Its Applications. Kluwer Academic Publishers, vol. 46, Dordrecht, Boston, London, 1989. · Zbl 0683.32002  Chern-Levine-Nirenberg. Intrinsec norms on a complex manifolds,Global Analysis, University of Tokyo Press, 119–139, 1969.  Demailly, J.P. Potential theory in sevrai complex variables, cours École d’été C.I.M.P.A, Nice, juillet 89.  Demailly, J.P. Monge-Ampère operator, Lelong numbers and intersection theory, Complex Analysis and Geometry Ancona, V. and Silva, A., Eds.,CIRM, Univ. de Trente, (1991).  EL Mir, H. Sur le prolongement des courants positifs fermés,Acta. Math.,153, 1–45, (1984). · Zbl 0557.32003  EL Mir, H. Extending positive currents with conditions on slices, Proc. of the International Conference on Several Complex Variables, Kyoto, 322–326, 1988.  EL Mir, H. et Amamou, M. Sur le prolongement des courants positifs fermés avec des conditions sur les tranches,CRAS,315, serie I, 777–780, (1992). · Zbl 0762.32005  Federer, H.Geometric Mesure Theory, Berlin, New York, Springer-Verlag, 1969. · Zbl 0176.00801  Fornaes, J.E. et Sibony, N. Oka’s inequality for currents and applications,Math. Ann.,301, 399–419, (1995). · Zbl 0832.32010  Harvey, R. and Shiftman, B. A. characterization of Holomorphic chains,Ann. Math.,99, 553–587, (1974). · Zbl 0287.32008  Harvey, R. and Polking, J. Extending of analytic objects,Comm. Pure Appl. Math.,28, 701–727, (1975). · Zbl 0323.32013  Kiselman, O.C. Sur la définition de l’opérateur de Monge-Ampère complexe, proceedingLectures Notes Math.,1094, Toulouse, 1983.  Lelong, P. Fonctions plurisouharmoniques et formes différentielles positives, Dunod, Paris, Gordon Beach, New York, 1968. · Zbl 0195.11603  Lelong, P. Sur la structure des courants positifs fermés, Seminaire P. Lelong 1975/76,Lectures Notes Math.,578, 136–156, Springer-Verlag, (1977).  Lelong, P. et Gruman, L. Entires fonctions of several complex variables,Grundlehren Math., Wiss. 282, Springer-Verlag, (1986). · Zbl 0583.32001  Oka, O. Sur les fonctions analytiques de plusieurs variables, IX. Une mode nouvelle engendrant les domaines pseudo-convexes,Jap. J. Math.,32, (1962). · Zbl 0138.06501  Raby, G. Tranchage des courants positifs fermés et équation de Lelong-Poincaré,J. Math. Pures App., (à paraître), 1995.  Riemenschneider, O. Uber den flacheneinhalt analytischen, mengen und die Erzengung k-pseudoconvexe Gebiete,Invent. Math.,2, 307–331, (1967). · Zbl 0148.32101  Schwartz, L.Théorie des Distributions, Herman, Paris, 1966.  Skoda, H. Nouvelle méthode pour l’étude des potentiels associés aux ensembles analytiques, Seminaire P. Lelong,Lectures Notes Math.,410, 117–141.  Skoda, H. Prolongement des courants positifs fermés, de masse finie,Inv. Math.,88, 361–376, (1982). · Zbl 0488.58002  Skoda, H. Extension problems and positive currents in complex analysis, Université Paris VI, 4 place Jussieu, Paris. · Zbl 0596.32020  Siu, Y.T. Extention of meromorphic maps into Kähler manifolds,Ann. Math.,102, 421–462, (1975). · Zbl 0318.32007  Siu, Y.T. Analyticity of sets associated to Lelong Numbers and the extension of closed positive currents,Inv. Math.,27, 53–156, (1974). · Zbl 0289.32003  Sibony, N. Quelques problèmes de prolongement de courants en analyse complexe,Duke Math. J.,52, 157–197, (1985). · Zbl 0578.32023  Stoll, W. Über die Fortsetzbarkeit analytischer Mengen endlichen Oberflücheninhaltes,Arch. Math.,9, 167–175, (1958). · Zbl 0083.30801
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