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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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