One of the fundamental analytic tools used in complex geometry is the Ohsawa-Takegoshi (O-T) extension theorem. For any particular application one typically needs a new version of this result. In this paper variants of O-T theorem are proven when \(h_L\) a Hermitian metric on a line bundle \(L\) over a projective manifold \(X\) admits singularities, and the sub-variety \(Y\) of \(X\), from which we extend canonical forms, is a divisor with simple normal crossings. Since the statements are rather lengthy here I describe the first main result referring to the paper for other (and also related conjectures). Thus apart from the objects mentioned above one considers \(h_Y\) a smooth metric on the bundle corresponding to \(\mathcal{O} (Y )\). The curvatures of the metrics satisfy \[ \Theta _{h_L} \geq 0, \ \ \ \Theta _{h_L} \geq \delta \Theta _{h_Y} (Y), \ \ \ \delta >0. \] Locally \(h_L =e^{-\varphi _L}, \ \ h_Y=e^{-\varphi _Y}\), and for some holomorphic functions \(f_j\), not identically zero on components of \(Y,\) \[ \varphi _L = \sum a_j \log |f_j |^2 +\tau _L , \] where \(a_j \) are positive numbers and \(\tau _L\) is non-singular.
For a twisted canonical form on \(Y\): \(u\in H^0 (X, (K_X +Y+L)\otimes \mathcal{O}_X / \mathcal{O}_X (-Y) )\) its restrictions to coordinate charts \(u_{|V_j}\) have extensions \(U_j \in H^0 (V_j , (K_X +Y+L)\) satisfying \[ \int_{V_j } |U_j |^2 e^{-\varphi _L -\varphi _Y} < \infty . \] Apart from those rather standard assumptions there are two more concerning singularities. In some open set \(V\) containing singularities of \(Y\) there is a snc divisor \(W\) given by \(\{ z_j =0\} , \ j\leq m,\) such that \[ \varphi _L = \sum (1-\frac{1}{k_j })\log |z_j |^2 +\tau _L , \] \(k_j\) positive integers, \(\tau _L\) bounded.
Moreover the curvature of the restriction of \(h_L\) to \(V\) is bounded from below by constant times the restriction of a fixed conic Kähler metric \(\omega _{\mathcal C}\) locally quasi-isometric with \[ \sum _1 ^m \frac{idz_j \wedge d\bar{z}_j }{|z_j|^{2-2/k_j } } + \sum _{j>m} idz_j \wedge d\bar{z}_j . \] Under the above hypothesis \(u\) extends to \(X\) and there exists a section \(U\in H^0 (X , (K_X +Y+L)\) with \(U_{|Y}=u\) satisfying the estimate \[ \int_{X\setminus V}|U|^2 e^{-\varphi _L -\varphi _Y} d \omega _{\mathcal C} \leq C\left[\int_{Y\setminus V} |\frac{u}{ds}|e^{-\varphi _L} + \sum_j \left(\int _{Y_j \cap V} |\frac{u}{ds}|_{ \omega _{\mathcal C} }^{2/(1+a)} e^{-\frac{\varphi _L}{1+a} } dV _{ \omega _{\mathcal C} }\right)^{1+a} \right], \] where \(s\) is the canonical section of \(\mathcal{O} (Y )\) normalized by \(|s|_{h_Y} \leq e^{-\delta}\) , \(a\in (0,1]\) is a prescribed number, and then \(C\) depends on \(a\), constant from curvature inequalities, geometry of the conic metric, and the upper bound on \(Tr_{ \omega _{\mathcal C} }dd^c \tau _L .\) The explicit dependence can be extracted from the proof.