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The \(\overline\partial\)-equation on a positive current. (English) Zbl 1031.32005
The authors study the induced \(\overline\partial\)-equation on a positive current in a complex manifold \(M\). More precisely, let \(u\) and \(f\) be smooth differential forms, \(T\) a positive current on \(M\). \(\overline\partial u=f\) on \(T\) means – by definition – that (*) \(\overline\partial u\wedge T=f\wedge T\). In the case when \(T=[V]\), the current of integration on an analytic set \(V\) in \(M\), (*) means that \(\overline \partial(u |_{V_{\text{reg}}})= f|_{V_{\text{reg}}}\) where \(V_{\text{reg}}\) denote the set of regular points of \(V\). The authors obtain a number of very interesting results, mostly for currents of bidimension (1,1). Let \(T\) be a positive current of bidimension \((p,p),\omega\) a smooth (1,1) form, \(\sigma_T=T \wedge\omega_p\) where \(\omega_p= {\omega^p \over p!}\) the trace measure of \(T\). For \(g\) a \((0,q)\) form in \(\mathbb{C}^n\), the \(L^2\)-norm of \(g\) over \(T\) is \(\|g\|^2= \int c_q g\wedge\overline g\wedge T\wedge \omega_{p-q}\) if \(T\) is of bidimension \((p,p)\); here \(c_q=(-1)^{q (q+1)/2} i^q=(-i)^{q^2}\) ensure that the \(c_qg \wedge \overline g\) is a positive form.
If \(f\) is a \((p,q)\) form, define \(\|f\|=\sup |\int f\wedge g\wedge T|\) the supremum being taken over all \(g(0,p-q)\) forms, with \(\|g\|<1\). Let \(L^2_{(p,q)}(T)\) be the completion of the space of \((p,q)\) forms with respect to the norms defined above.
Another definition is that of \(\Gamma\)-directed currents of bidimension \((p,p)\). For \(M\) a complex manifold with a Hermitian metric \(g\) and \(\omega\) the associated \((1,1)\) form, for \(X\subset M\) assume that for every \(z\in X\) then is given a cone \(\Gamma_z\) of positive currents of bidimension \((p,p)\), \(\varepsilon_Z\) supported at \(z\). Let \(\Gamma=\cup_{z\in X\gamma_z}\), assumed to be closed. Then \(T\) is \(\Gamma\)-directed if \(T\) belongs to the convex hull of \(\Gamma\); \(T\) is said to be \(\alpha\)-directed when \(\Gamma\) is defined by the Dirac currents orthogonal to the family of differential forms \(\alpha\).
The authors show, that under some topological condition, if \(X\) is a compact set in a complex Hermitian manifold \((M,\omega)\) then there exists a positive current \(T\), supported on \(X\) of mass 1, of bidimension \((1,1)\), \(\Gamma\)-directed and such that \(i\partial\overline \partial T=0\), where \(\Gamma= U\Gamma_z\), with \(\Gamma_z\) a cone of Dirac currents of bidimension \((1,1)\) supported in \(z\).
The authors, using Choquet’s theorem, study the local structure of currents directed by a foliation (near a regular point). If \(T\) is directed by \(dz_p,\dots, dz_n\) and if \(V_c=\{z\mid z_{p+1}= c_{p+1},\dots, x_n=C_n\}\) then \(T=\int T_c d\mu(c_{p+1},\dots,c_n)\) where \(\mu\) is a positive measure and \(\{T_c \}\) is a measurable family of positive currents (of bidimension \((p,p))\) supported on \(V_c\). If \(\partial\overline \partial T\leq 0\) then \(T_c=u_c[V_c]\) where \([V_c]\) is current of integration on \(Vc\), \(u_c\) non-negative plurisuperharmonic function on \(V_c\), and if \(i\partial\overline \partial T=0\), \(u_c\) is pluriharmonic.
When \(T\) is a positive non-closed current, the authors introduce the notion of \(L^2\)-normal current, i.e. satisfying \(dT=\tau \wedge T\) with \(\tau\) a 1-form in \(L^2(T)\), or equivalently that \(\partial T=e_1\wedge T\) with \(e_1\) a \((1,0)\) form in \(L^2_{\text{loc}} (T)\). Under some condition on \(M\), and \(\dim_\mathbb{C} M>2\), if \(T\) is a smooth, strictly positive \(L^2\)-normal \((1,1)\) form then there exists a function \(\varphi\) such that \(e^{-\varphi}T\) is closed (for \(\dim_\mathbb{C} M=2\), every strictly positive \((1,1)\) form on \(M\) is \(L^2\)-normal).
There are many examples of \(L^2\)-normal forms that are not proportional to a closed form, and the authors give characterization of currents satisfying \(\partial T=\tau\wedge T\) \((T\) of bidimension \((1,1)\) in \(D\subset\mathbb{C}^n)\). In particular if \(T\) (of bidimension \((p,p)\) in \(D\subset\mathbb{C}^n\)) is directed by a holomorphic foliation of dimension \(p\) and if \(i\partial\overline\partial T\wedge\omega^{\prime 1}=0\) then \(\partial T=\tau \wedge T\).
The authors define the notions of strongly holomorphic (\(T\)-holomorphic), respectively weakly holomorphic function on \(T\) \((\omega\)-holomorphic) \(u\in L^2_{\text{loc}} (\sigma_T)\) \((T\) positive curent of dimension \((p,p)\) on an open set \(D\subset \mathbb{C}^n)\) is \(T\)-holomorphic if \((\exists)\) \(\{u_n\}\) of \(C^2\)-functions such that \(u_n\) tends to \(u\) in \(L^2_{\text{loc}} (\sigma_T)\) and \(\overline \partial _n\to 0\) in \(L^2_{\text{loc}}(T)\). If \(\partial T=\tau\wedge T\) with \(e\in L^2_{\text{loc}} (\sigma_T)\), \(u\in L^2_{\text{loc}} (\sigma_T)\) is \(\omega\)-holomorphic if \(\overline \partial u\wedge T\) \((=\) by definition to \(\overline\partial (u\wedge T)-u\overline \tau\wedge T)\) is zero in the sense of currents. In the same spirit they define \(T\)-plurisubharmonic functions.
Then: if \(T\) is positive, closed, of bidimension \((p,p)\) in \(D\), \(u\) \(T\)-holomorphic implies that \(Du \in L^2_{\text{loc}} (T)\), \(i\partial \overline\partial |u|^2 \wedge T=i \partial u\wedge\overline \partial u\wedge T\) (and in particular \(|u |^2\) is \(\omega\)-plurisubharmonic), and we have estimates on compact sets in \(D\). From these estimates one deduces that no strongly holomorphic function with respect to \(T\), \(T\) positive, closed, of bidimension \((p,p)\) can have compact support in \(D\); on the other hand, the authors give a simple example of a positive, closed current \(T\) in \(\mathbb{C}^2\) which carries an \(\omega\)-holomorphic function with compact support. They also prove a version of Riemann extension for a \(T\)-holomorphic function, \(T\) being a positive current of bidimension \((p,p)\) such that \(i\partial\overline\partial T=0\).
The main results of the paper concern the \(\overline\partial\)-equation on currents of bidimension \((1,1)\). Let \(D\) be an open set in \(\mathbb{C}^n\), \(T\) a positive current of bidimension \((1,1)\), \(L^2\)-normal and \(i\partial \overline\partial T\leq 0\). Then for any \((1,1)\) form \(f\) in \(L^2_{\text{loc}}(T)\) there exists a \((1,0)\) form \(u\in L^2(T)\) which satisfies (*).
If \(\varphi\) is a smooth plurisubharmonic function on \(D\), this solution satisfies \(\|u\|_\varphi \leq\|f \|_\varphi\). The same result is proved if, instead of an open set in \(\mathbb{C}^n\), one considers a Hermitian manifold \((M,\omega)\). If \(T\) is a current on \(D\) as before, and if \(\partial T=\tau \wedge T\) with \(\tau\in L^2_{\text{loc}}(T)\), \(\varphi\) a strictly plurisubharmonic function on \(D\) and \(\tfrac 12 i\partial \overline\partial_\varphi\geq \varepsilon i\geq\wedge\overline \tau+ \omega\) for some \(\varepsilon >0\), then for a \((0,1)\) form \(f\) on \(T\) such that \(\int i\overline f\wedge f\wedge Te^\varphi<\infty\) the equation (*) has a solution \(u\) (verifying an integral estimate) iff \(\int f\wedge h\wedge T=0\) for all \((1,0)\) forms \(h\) such that \(\overline\partial (h\wedge T)=0\), \(\|h \|_\varphi <\infty\). In particular, the image of \(\overline\partial\) is closed.
As an application of these results, the authors construct meromorphic functions on lamination by Riemann surfaces. After some linear algebra on \(L^2\)-spaces on currents of bidegree \((1,1)\) and some considerations on weak and strong extension of \(\overline\partial\) and their adjoint, they obtain, generalising the Kodaira-Nakani-Hörmander identity, a priori estimates for the solutions of the \(\overline\partial\)-equation, existence theorems for \(\overline\partial\) on closed currents of bidegree \((1,1)\). These results are extended to some compact Kähler manifolds. The paper contains two Appendices, the second gives a concrete interpretation in the case of currents defined by analytic varieties. It is shown that one of the existence theorems contains, as a particular case, a key result on local solvability of \(\overline\partial\) of W. L. Pardon and M. A. Stern [J. Am. Math. Soc. 4, 603-621 (1991; Zbl 0751.14011)], in the case the variety is of codimension one. This results is also generalized to arbitrary codimension and allows to replace the analytic variety by a lamination by complex manifolds.

MSC:
32C30 Integration on analytic sets and spaces, currents
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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