A Hartogs type extension theorem for generalized \((N,k)\)-crosses with pluripolar singularities (Q2905528)

The author discusses an extension theorem for separately holomorphic functions with pluripolar singularities in the following general context. Let \(D_1,\dots,D_N\) (\(N\geq2\)) be Riemann domains and let \(A_j\subset D_j\) be locally pluriregular, \(j=1,\dots,N\). Fix a \(k\in\{1,\dots,N\}\) and let \(\mathcal T:=\{\alpha\in\{0,1\}^k: |\alpha|=k\}\). For \(\alpha\in\mathcal T\) with \(\alpha_{i_1}=\dots=\alpha_{i_k}=1\), \(i_1<\dots<i_k\), \(\alpha_{j_1}=\dots=\alpha_{N-k}=0\), \(j_1<\dots<j_{N-k}\), put \(D^\alpha:=D_{i_1}\times\dots\times D_{i_k}\), \(A^\alpha:=A_{j_1}\times\dots\times A_{j_{N-k}}\).NEWLINENEWLINENEWLINENEWLINE Suppose that we are given pluripolar sets \(\Sigma_\alpha\subset A^\alpha\), \(\alpha\in\mathcal T\). Put \(\chi_\alpha:=A^\alpha\times D^\alpha\), \(\chi_\alpha^\Sigma:=(A^\alpha\setminus\Sigma_\alpha)\times D^\alpha\). Define the \textit{\((N,k)\)-cross} \(\mathbf{X}:=\bigcup_{\alpha\in\mathcal T}\chi_\alpha\) and the \textit{generalized \((N,k)\)-cross} \(\mathbf{T}:=\bigcup_{\alpha\in\mathcal T}\chi_\alpha^\Sigma\).NEWLINENEWLINENEWLINENEWLINE Let further \(\widehat{\mathbf{X}}:=\big\{(z_1,\dots,z_N)\in D_1\times\dots\times D_N: h_{A_1,D_1}(z_1)+\dots+h_{A_N,D_N}(z_N)<1\big\}\), where \(h_{A_j,D_j}\) denotes the relative extremal function. Let \(M\subset\mathbf{T}\) be a relatively closed pluripolar set such that for every \(a\in A^\alpha\setminus\Sigma_\alpha\) the fiber \(M_{a,\alpha}:=\{z\in D^\alpha: (a,z)\in M\}\) is pluripolar. We say that a function \(f:\mathbf{T}\longrightarrow\mathbb C\) is \textit{separately holomorphic} (\(f\in\mathcal O_S(\mathbf{T}\setminus M)\)) if for every fixed \(a\in A^\alpha\setminus \Sigma_\alpha\) the function \(z\longmapsto f(a,z)\) is holomorphic in the domain \(D^\alpha\setminus M_{a,\alpha}\). Let \(\mathcal O_S^c(\mathbf{T}\setminus M)\) denote the set of all \(f\in\mathcal O_S(\mathbf{T}\setminus M)\) such that for every fixed \(b\in D^\alpha\) the function \(z\longmapsto f(z,b)\) is continuous on the set \(\{z\in A^\alpha\setminus\Sigma_\alpha: (z,b)\notin M\}\).NEWLINENEWLINENEWLINENEWLINE Finally, let \(\mathcal F:=\mathcal O_S(\mathbf{X}\setminus M)\) if \(\mathbf{T}=\mathbf{X}\) and \(\mathcal F:=\mathcal O_S^c(\mathbf{T}\setminus M)\) otherwise. NEWLINENEWLINENEWLINENEWLINE The author proves that in above situation there exist a relatively closed pluripolar set \(\widehat M\subset\widehat{\mathbf{X}}\) and a generalized \((N,k)\)-cross \(\mathbf{T}'\subset\mathbf{T}\) such that \(\widehat M\cap\mathbf{T}'\subset M\) and every function \(f\in\mathcal F\) extends holomorphically to \(\widehat{\mathbf{X}}\setminus\widehat M\).