An injectivity theorem (Q2877511)

Let \((X, B)\) a proper log variety and \(L\) a Cartier divisor on \(X\) such that \(L\sim_{{\mathbb R}} K_X + B\). In the paper under review, the author answers affirmatively the lifting problem which asks whether the restriction map NEWLINE\[NEWLINE \Gamma(X, \mathcal{O}_X(L)) \longrightarrow \Gamma(Y, \mathcal{O}_Y(L))NEWLINE\]NEWLINE is surjective for a closed subvariety \(Y\subset X\), in the case that \(Y\) is the locus of non-log canonical singularities of \((X,B)\), if it is non-empty (Theorem~6.2).NEWLINENEWLINEIn order to prove this, the author extends a theorem by \textit{H. Esnault} and \textit{E. Viehweg} [Lect. Notes Math. 947, 241--250 (1982; Zbl 0493.14012); Lectures on vanishing theorems. Notes, grew out of the DMV-seminar on algebraic geometry, held at Reisensburg, October 13-19, 1991. Basel: Birkhäuser Verlag (1992; Zbl 0779.14003)], which states that, for a proper non-singular variety \(X\) and a Cartier divisor \(L\) such that \(L\sim_{\mathbb Q} K_X + \sum_i b_iE_i\), where \(\sum_i E_i\) a normal crossing divisor and \(b_i\in{\mathbb Q}\) with \(0\leq b_i\leq 1\), and an effective divisor \(D \subset \text{Supp}(\sum_{0<b_i<1}E_i)\), the homomorphism NEWLINE\[NEWLINE H^q(X, \mathcal{O}_X(L)) \longrightarrow H^q(X, \mathcal{O}_X(L+D))NEWLINE\]NEWLINE is injective for all \(q\), to the case of \(D \subset \text{Supp}(\sum_{0<b_i\leq 1}E_i)\) (Theorem~2.3).NEWLINENEWLINEAs applications, it is shown that (1) for \((X, B)\) a proper connected log variety such that \(K_X + B \sim_{\mathbb R} 0\), if the locus \(Y\) of non-log canonical singularities is non-empty, then \(Y\) is connected and intersects every lc center of \((X,B)\) (Global inversion of adjunction, Theorem~6.3), and (2) assume that \((X, B)\) is a proper log variety with log canonical singularities and \(L\) is a Cartier divisor on \(X\) such that \(H = L -(K_X + B)\) is a semiample \({\mathbb Q}\)-divisor. If the union \(Z\) of lc centers of \((X,B)\) is contained in \(\text{Supp}(D)\) for a divisor \(D\in | m_0 H|\) with \(m_0\geq 1\), then the restriction homomorphism NEWLINE\[NEWLINE \Gamma(X, \mathcal{O}_X(L)) \longrightarrow \Gamma(Z, \mathcal{O}_Z(L))NEWLINE\]NEWLINE is surjective (extension from lc centers, Theorem~6.4).