Quantitative extensions of pluricanonical forms and closed positive currents (Q2880836)

Let \(X \rightarrow \Delta\) be a smooth projective family of complex manifolds over the unit disc, and let \((F, h_F)\) be a line bundle over \(X\) such that the curvature current is semipositive and the restriction of the metric \(h_F\) to the central fibre \(X_0\) is well-defined. Under this hypothesis the classical extension theorem of \textit{T. Ohsawa} and \textit{K. Takegoshi} [``On the extension of \(L^2\) holomorphic functions'', Math. Z. 195, 197--204 (1987; Zbl 0625.32011)] allows to extend sections of \((K_X+L)|_{X_0}\) which are \(L^2\) with respect to \(h_F\) to the total space \(X\). Moreover we have an estimate for the extension depending on a universal constant. The aim of this paper is to prove similar extension statements for bundles of type \(p K_X+qL\) where \(p\) and \(q\) are positive integers. If \(L\) is trivial this problem has been settled by \textit{Y.-T. Siu} in his work on invariance of plurigenera [``Invariance of plurigenera'', Invent. Math. 134, No. 3, 661--673 (1998; Zbl 0955.32017); ``Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type'', in: Complex geometry. Berlin: Springer. 223--277 (2002; Zbl 1007.32010)]. \newline For many applications in algebraic geometry it is however crucial to understand the general case. Examples show that it is not sufficient to replace the \(L^2\) condition in the Ohsawa-Takegoshi theorem by an \(L^{2/p}\)-condition. In order to resolve this problem the authors consider the ideal \(\overline{\mathcal I_0^{(q)}}\), that is the integral closure of the \(q\)-th power of the multiplier ideal \(\mathcal I(h_L|_{X_0})\) associated to \(h_L|_{X_0}\). They prove that there exists a universal constant \(C_0>0\) such that for any pair of positive integers \(p \geq q\) and for any section NEWLINE\[NEWLINE u \in H^0\Big(X_0, (pK_{X}+ qL)\big|_{X_0} \otimes \overline{\mathcal I_0^{(q)}}\,\Big) NEWLINE\]NEWLINE there exists a section NEWLINE\[NEWLINE U \in H^0\big(X, pK_{X}+ qL\big) NEWLINE\]NEWLINE such that over the central fiber we have \(U\big|_{X_0}= u \otimes d\pi^{\otimes p}\) and the following \(L^{2/p}\) integrability condition holds: NEWLINE\[NEWLINE \int_{X} |U|^{2\over p} e^{-{q\over p}\varphi_L}\leq C_0 \int_{X_0} |u|^{2\over p} e^{-{q\over p}\varphi_L}. NEWLINE\]NEWLINE Using the same strategy of proof the authors establish the following geometric version of their theorem: Let \(X\) be a projective manifold, and let \((L, h_L)\) be a hermitian line bundle on \(X\). Let \(S\subset X\) be a non-singular, irreducible submanifold of codimension 1, such that the restriction of \(h_L\) to \(S\) is well defined. Assume that the curvature condition \(\Theta_{h_L}(L)\geq \varepsilon_0\omega\) is satisfied. Then any section of \((pK_S+ qL)\otimes \overline{\mathcal I(h_L|_S)^{(q)}}\) extends to \(X\) as a section of the line bundle \(p(K_X+ S) +qL\). NEWLINENEWLINENEWLINENEWLINE The second part of the paper deals with improvements of these statements that aim to replace the ideals \(\overline{\mathcal I_0^{(q)}}\) by a more accessible condition. Results of this type play an important role in \textit{C. D. Hacon} and \textit{J. McKernan}'s work on existence of flips [``Existence of minimal models for varieties of log general type. II'', J. Am. Math. Soc. 23, No. 2, 469--490 (2010; Zbl 1210.14021)], we refer to the very interesting introduction for the precise statements.