Skoda division of line bundle sections and pseudo-division (Q2810651)

The author presents modifications and generalizations of Skoda-type division theorems for holomorphic sections of line bundles over a smooth projective variety \(X\). NEWLINEHe is interested in an alternative approach to the finite generation of the canonical ring of \(X\) and discusses the meaning of his results for this project. NEWLINELet \(G\) and \(H \) be holomorphic line bundles on \(X\) and \(g_1,\dots,g_r\in H^0 (X,G)\). NEWLINEThe following Skoda-type division theorem is very general for the above setting and essential for the other results of the paper: Let \((G,e^{-\eta})\) and \((H,e^{-\psi})\) be singular hermitian metrics with nonnegative curvature currents for \(G\) and \(H\) respectively. NEWLINELet \(q\geq\min(\dim X,p-1)\), \(q\in\mathbb N\). NEWLINEIf \(f\in H^0 (X,K_X\otimes H\otimes (q+2)G))\) satisfies \(\int_X|f|^2\left(\frac{1}{|g_1|^2+\dots+|g_r|^2}\right)^{(q+1)}e^{-\eta}e^{-\psi}<\infty\), then there exist \(h_1,\dots,h_r\in H^0 (X,K_X\otimes H\otimes (q+1)G)\) such that \(f=g_1h_1+\dots +g_rh_r\). NEWLINEThe proof can be achieved by combining results of several authors, e.g., [\textit{D. Varolin}, Math. Z. 259, No. 1, 1--20 (2008; Zbl 1140.32006)], [\textit{J.-P. Demailly}, in: School on vanishing theorems and effective results in algebraic geometry. NEWLINELecture notes of the school held in Trieste, Italy, April 25--May 12, 2000. Trieste: The Abdus Salam International Centre for Theoretical Physics. 1--148 (2001; Zbl 1102.14300)], [\textit{Q. Guan} and \textit{X. Zhou}, Ann. Math. (2) 182, No. 2, 605--616 (2015; Zbl 1329.32016)]. NEWLINEThe author introduces the notion of pseudo-division for general holomorphic line bundles \(L\), \(G\) over \(X\) and \(f\in H^0(X,L\otimes G)\), \(g_1,\dots,g_r\in H^0 (X,G)\). If there exist a line bundle \(W\) on \(X\) and non-zero holomorphic sections \(w_1,\dots,w_t\in H^0 (X,W) \) and sections \(h_{ijk}\in H^0 (X,W)\), \(1\leq i\leq t\), \(1\leq j\leq r\), \(1\leq k\leq t\), satisfying \(fw_i=g_1H_{i1}+\dots+g_rH_{ir}\) with \(H_{ij}=\sum_{k=1}^t w_kh_{ijk}\), then \(f\) is said to be pseudo-divided by \(g_1,\dots,g_r\). NEWLINEThe main result is as follows. NEWLINEAssume that \(G\) is ample and \(Z\) is a smooth subvariety of \(X\) with associated ideal sheaf \(I_Z\). NEWLINEFor every finite generating system \(\{g_1,\dots,g_r\}\) of sections of the sheaf \(G\otimes I_Z\) there exists \(m_0\geq 1\) with the following property: If \(m\geq m_0\) and \(f\in H^0 (X,K_X\otimes mG)\) vanishes along \(Z\), then \(f\) is pseudo-divided by \(g_1,\dots,g_r\). NEWLINEIn connection with the finite generation problem for graded rings the author shows by example that basic statements may remain true when division conditions are replaced by pseudo-division conditions.