A transcendental approach to Kollár's injectivity theorem. II (Q2848379)

The author proves a relative version of the injectivity theorem in [\textit{O. Fujino}, Osaka J. Math. 49, No. 3, 833--852 (2012; Zbl 1270.32004)] and it is a generalization of \textit{J. Kollár}'s injectivity theorem [Ann.Math. (2) 123, 11--42 (1986; Zbl 0598.14015)].NEWLINENEWLINELet \(f:X\rightarrow Y\) be a proper surjective morphism from a Kähler manifold X to a complex variety \(Y\) and \((E,h_E)\) and \((L,h_L)\) be a holomorphic vector bundle and a holomorphic line bundle on \(X\) with smooth hermitian metrics; suppose that there exists a holomorphic line bundle \(F\) on \(X\) with a singular hermitian metric \(h_F\) such that:NEWLINENEWLINE(i) there exists \(Z\subset X\) such that \(h_F\) is smooth on \(X\setminus Z\);NEWLINENEWLINE(ii) \(\sqrt{-1}\Theta(F)\geq -\gamma\) for \(\gamma\) a smooth \((1,1)\)-form on \(X\);NEWLINENEWLINE(iii) \(\sqrt{-1}(\Theta(E)+\mathrm{Id}_E\otimes\Theta(F))\) is Nakano semi-positive on \(X\setminus Z\);NEWLINENEWLINE(iv) \(\sqrt{-1}(\Theta(E)+\mathrm{Id}_E\otimes\Theta(F)-\epsilon \mathrm{Id}_E\otimes\Theta(L))\) is Nakano semi-positive on \(X\setminus Z\) for some \(\epsilon>0\).NEWLINENEWLINEThen, for \(s\) a nonzero holomorphic section of \(L\), the multiplication homomorphism NEWLINE\[NEWLINER^qf_*(K_X\otimes E\otimes F\otimes\mathcal{I}(h_F))\rightarrow R^qf_*(X,K_X\otimes E\otimes F\otimes \mathcal{I}(h_F)\otimes L)NEWLINE\]NEWLINE is injective for \(q\geq 0\), where \(\mathcal{I}(h_F)\) is the multiplier ideal associated to \(h_F\).NEWLINENEWLINEThe proof is illustrated in Section 3. Section 4 then contains applications of the main results to injectivity and vanishing theorems in algebraic geometry, such as a Kawamata-Viehweg-Nadel type and a Kollár type vanishing theorems.