Pfaff systems, currents and hulls
In the first part of the article, the author shows the following. Let \(M\) be a compact complex Hermitian manifold of dimension \(k\). Let \((\alpha_1,\ldots,\alpha_{k-1})\) be continuous \((1,0)\)-forms on \(M\) (a Pfaff system). Then there exists a positive current \(T\) on \(M\) of mass 1 and bidimension \((1,1)\) such that \(T\wedge\alpha_j=0\) for all \(j=1,\ldots,k-1\) and \(\partial\bar\partial T=0\), without any integrability assumption on the \(\alpha_j\). In the second part of the article, the author studies the pluripotential theory of functions in a cone \(\mathcal{P}_{\Gamma}\). Informally, this is the class of smooth functions in \(\mathbb{C}^k\) for which subharmonicity is prescribed only in certain directions (given by \(\Gamma\)). One can define the hull of a compact set \(K\), \[ \hat K_{\Gamma} := \Bigl\{ z\in\mathbb{C}^k: u(z)\leq \sup_K u \text{ for every }u\in\mathcal{P}_{\Gamma} \bigr\}, \] as well as the class of positive \(\Gamma\)-directed currents of bidimension \((1,1)\), \[ \mathcal{P}_{\Gamma}^0:= \{T\geq 0, \text{ of bidimension } (1,1), T\wedge i\partial\bar\partial u\geq 0 \text{ for every } u\in\mathcal{P}_{\Gamma} \}. \] The above notions, as well as Jensen measures (which are an important tool), enter into most of the main results. These include the characterization of a Perron-Bremmerman type upper envelope of functions in \(\mathcal{P}_{\Gamma}\) in terms of an asymptotic notion of harmonicity. The problem of extending positive \(\Gamma\)-directed currents through `small' sets is also covered; to this end, the notion of \(\mathcal{P}_{\Gamma}\)-pluripolar sets is required.
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