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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.

32U40 Currents
32C30 Integration on analytic sets and spaces, currents
32U25 Lelong numbers
32W20 Complex Monge-Ampère operators
Full Text: DOI
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