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Problème de Plateau complexe dans les variétés kählériennes. (Complex Plateau problem in {K}ähler manifolds). (French) Zbl 1017.32013

The complex Plateau problem (or the bord problem) in a complex manifold \(X\) is the problem of characterizing the real submanifolds \(\Gamma\subset X\) which are boundaries (in the sense of currents) of analytic subvarieties of \(X\setminus\Gamma\). A complex manifold \(\omega\) is called disk-convex if for any compact \(K\subset\omega\) there exists a compact \(\hat K\subset\omega\) which contains any irreducible analytic curve \(A\) in \(\omega\setminus K\) such that \(A\cup K\) is compact and \(\bar A\cap K\neq\emptyset\).
The main result of the paper can be formulated as follows. Consider a manifold \(X = U\times\omega\), where \(U\) is connected of dimension \(n\) and \(\omega\) a disk-convex Kähler manifold, denoting by \(\pi: X\to U\) the projection, and let \(\Gamma\subset X\) be a real submanifold of dimension \(2n+1\) which is maximally complex and such that \(\Gamma\cap\pi^{-1}(K)\) is compact for any compact \(K\subset U\). Denote \(\gamma_z = \Gamma\cap(\{z\}\times\omega), z\in U\). Let us suppose that each \(\gamma_z\) is a piecewise smooth curve with finitely many smooth pieces, and that \(\Gamma\) is transversal to \(\{z\}\times\omega\) at any point of \(\gamma_z\) except for a finite number of points. Let \([\gamma_z]\) denote the current of integration over \(\gamma_z\).
Then the following assertions are equivalent:
1) For each \(z\) from a \((n-1)\)-generic subset of U, there exists a holomorphic 1-cochain \([S_z]\) of a finite mass in \((\{z\}\times\omega)\setminus\gamma_z\) such that \(\bar S_z\) is compact and \([\gamma_z] = d[S_z]\).
2) There exists a non-empty open set \(O\subset U\) such that \(\Gamma\) admits a solution of the bord problem in \(O\times\omega\).
3) Each point \(z\in U\), except of a subset of measure zero (of dimension \(2n-1\)), possesses a neighborhood \(O_z\subset U\) such that \(\Gamma\) admits a solution of the bord problem in \(O_z\times\omega\).
4) \(\Gamma\) admits a solution of the bord problem in \((U\setminus Y)\times\omega\), where \(Y\subset U\) is a closed subset of measure zero (of dimension \(2n\)).
As applications of his main result, the author obtains the results of F. Reese Harvey and H. Blaine Lawson jun. [Ann. Math. 106, 213-238 (1977; Zbl 0361.32010)], P. Dolbeault and G. Henkin [Bull. Soc. Math. Fr. 125, 383-445 (1997; Zbl 0942.32007)] and T.-C. Dinh [Ann. Inst. Fourier 48, 1483-1512 (1998; Zbl 0916.32011)]. A generalization of the main theorem to non-smooth currents \(\Gamma\) is also formulated and a proof is outlined.

MSC:

32D15 Continuation of analytic objects in several complex variables
32C30 Integration on analytic sets and spaces, currents
32V40 Real submanifolds in complex manifolds
32V05 CR structures, CR operators, and generalizations
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