Ladders on Fano varieties (Q1296200)

The author proves the following very applicable theorem: Let \(H\) be a nef and big Cartier divisor on an \(n\)-dimensional normal variety \(X\) such that \((X, B_X)\) is Kawamata log terminal for some boundary divisor \(B_X\) and \(-(K_X + B_X) \equiv (n-r+1) H\) for some rational number \(r\) satisfying \(r > 0\) and \(n-r+1> 0\); then \(\dim|H|\geq n-1\), \(|H|\) does not have fixed components and \((X, B_X+S)\) is pure log terminal for a general \(S \in |H|\). The conditions are all satisfied by every log terminal Fano 3-fold \(X\) (taking \(B_X = 0\)). This theorem enables us to construct a ladder of log terminal Fano varieties with decreasing dimensions, and reduces problems to lower dimension case (e.g. calculation of \(\pi_1\) of the smooth part of \(X\)). A similar result might be true for an arbitrary log terminal Fano 4-fold according to the author's recent comments.