Some inequalities for minimal fibrations of surfaces of general type over curves (Q1801842)

Let \(g:Y\to C\) be a surjective morphism from a smooth complex projective 3-fold onto a smooth curve, whose general fibers are irreducible surface of general type. Applying divisorial contractions and flips, one can modify \(Y\) into a normal projective \(\mathbb{Q}\)-factorial 3-fold \(X\) with only terminal singularities so that \(g\) induces a morphism \(f:X \to C\) and \(K_ X\) is \(f\)-nef; such a fibration is called a (relatively) minimal fibration of surfaces of general type over \(C\). Then \(K^ 3_ X\) is a well-defined rational number determined independently of the choice of a minimal fibration \(f\). The author then gives an inequality bounding \(K^ 3_ X\) from below in terms of \(p_ g(F)\), \(K^ 2_ F\), \(\chi( {\mathcal O}_ F)\), \(\chi({\mathcal O}_ X)\) and the genus \(b\) of \(C\), where \(F\) is a general fiber of \(f\). The inequality is sharp in the sense that if the equality holds the fibration becomes isotrivial, i.e., two general fibers are isomorphic to each other. Thus, the author extends the results in the surface case to the case of fibered 3-folds.