Inductive approach to effective b-semiampleness (Q2878715)

Let \((X, B)\) be a pair of data with a normal variety \(X\) and a Weil divisor \(B\) such that \(K_X+ B\) is \(\mathbb{Q}\)-Cartier. An lc-trivial fibration is the data of a pair \((X, B)\) and a fibration such that \((K_X+B)|_F\) is torsion where \(F\) is the general fiber. \(K_X+B\) satisfies the equation NEWLINE\[NEWLINE K_X+ B+\frac{1}{r}(\phi)= f^*(K_Z+ B_Z + M_Z), NEWLINE\]NEWLINE where \(\phi\) is a rational function and \(r\) is the minimum integer such that \(r(K_X+B)|_F\sim 0\) (linearly equivalent to \(0\)). The moduli part \(M_Z\) is a \(\mathbb{Q}\)-Cartier divisor which is nef on some birational modification of \(Z\).NEWLINENEWLINEIn this paper, the author considers the the following conjecture due to \textit{Yu. G. Prokhorov} and \textit{V. V. Shokurov} [J. Algebr. Geom. 18, No. 1, 151--199 (2009; Zbl 1159.14020)].NEWLINENEWLINE\textbf{EbS(k)} (Effective b-semiampleness) For any lc-trivial fibration \(f: (X, B)\rightarrow Z\) with dimension of the generic fiber \(F\) equal to \(d\), dimension of \(Z\) equal to \(k\), and Cartier index of \((F, B|_F)\) equal to \(r\), there exist an integer \(m=m(d, r)\) and a birational morphism \(\nu: Z'\rightarrow Z\) such that \(mM_{Z'}\) is base point free.NEWLINENEWLINEThe author proves three theorems. First, by induction on the dimension of the base of the lc-trivial fibration, the author proves that \textbf{EbS(1)} implies \textbf{EbS(k)}.NEWLINENEWLINEBy the techniques of the theory of variations in Hodge structures, the author proves the second theorem:NEWLINENEWLINEFor any klt-trivial fibration \(f: (X, B)\rightarrow Z\) with \(M_Z\equiv 0\) (numerically equivalent to \(0\)) and \(\text{ Betti}_{\text{dim E'}}(E')=b\), where \(E'\) is a nonsingular model of the cover \(E\rightarrow F \) associated to the unique element of \(|r(K_F+B|_F)|\), there is an integer \(m=m(b)\) such that \(mM_Z\sim {\mathcal{O}}_Z\).NEWLINENEWLINEFinally, modifying the proof of the second theorem and applying more general setting in variation of mixed Hodge structures, the author proves the third theorem:NEWLINENEWLINELet \(f: (X, B)\rightarrow Z\) be an lc-trivial fibration with \(M_Z\equiv 0\), then \(M_Z\) is torsion.