\(J\)-pluripolar subsets and currents on almost complex manifolds
The main results of the paper are the following two theorems. (I) Let \(\Omega\subset\mathbb R^{2n}\) be an open set equipped with an almost complex structure \(J\) of class \(\mathcal C^1\). Let \(Y\subset\Omega\) be a \(\mathcal C^2\)-submanifold of codimension \(2p\) such that \(J(TY)=TY\). Then for each \(x_0\in Y\) there exist an open neighborhood \(U\) and \(f=(f_1,\dots,f_p)\in\mathcal C^2(U)\) such that \(Y\cap U=\{x\in U: f(x)=0,\; \overline\partial_Jf=0 \text{ on } Y\cap U,\;\partial_Jf_1\wedge\dots\wedge\partial_Jf_p\neq0 \text{ on } U\}\). Moreover, there exists a \(u\in\mathcal{PSH}(U,J)\cap\mathcal C^2(U\setminus Y)\) such that \(Y\cap U=u^{-1}(-\infty)\). The result was first proved by \textit{J. P.~Rosay} [Math. Z. 253, No.~4, 659--665 (2006; Zbl 1121.32012)] under more restrictive assumptions. (II) Let \((M,J)\) be an almost complex manifold equipped with a Hermitian metric \(\omega\). Let \(A\subset M\) be a closed subset such that there exists a \(u\in\mathcal{PSH}(M,J)\cap\mathcal C^2(M\setminus A)\) with \(A=u^{-1}(-\infty)\). Let \(T\) be a positive current of bidimension \((p,p)\) on \(M\setminus A\). Assume that the currents \(T\) and \(i\partial_J\overline\partial_JT\) have extensions \(\widetilde T\), \(\widetilde{i\partial_J\overline\partial_JT}\) on \(M\). Then: (1) For any \(u\in\mathcal{PSH}(M,J)\cap\mathcal C^2(M\setminus A)\) with \(A=u^{-1}(-\infty)\), compact set \(K\subset M\), and \(\alpha>0\) we have \(\int_{K\setminus A}T\wedge\omega^{p-1}\wedge\frac{i\partial_Ju\wedge\overline\partial_Ju}{u^2(\log-u)^{1+\alpha}}<\infty\). (2) There exists a positive current \(S\) supported by \(A\) such that \(\widetilde{i\partial_J\overline\partial_JT}=i\partial_J\overline\partial_J\widetilde T+S\). (3) There exists a sequence \((\rho_m)_{m=1}^\infty\) of smooth functions vanishing in a neighborhood of \(A\) and converging uniformly on every compact subset of \(M\setminus A\) to \(\mathbf{1}_{M\setminus A}\) such that \(d\widetilde T=\lim_{m\to+\infty}\rho_mdT\). The theorem generalizes some results of \textit{H.~El Mir} [Acta Math. 153, 1--45 (1984; Zbl 0557.32003)] and \textit{K.~Dabbek, F.~Elkhadhra} and \textit{E. M.~Hassine} [Math. Z. 245, No.~3, 455--481 (2003; Zbl 1046.32009)]. Moreover, the author presents a generalization of Federer and Bassanelli results for flat and \(\mathbb C\)-flat currents in the almost complex case.
scientific article; zbMATH DE number 5671546
scientific article