Geometric height inequalities (Q5961484)

Let \(B\) be a complex projective curve of genus \(q\), let \(f: X\to B\) be a fibration of a smooth complex projective variety \(X\) over \(B\), let \(g\) denote the genus of the generic fiber of \(f\), let \(s\) denote the number of singular fibers of \(f\), and let \(\omega_{X/B}\) be the relative dualizing sheaf. Let \(C_1,\dots,C_n\) be mutually disjoint sections of \(f\), and let \(D\) be the divisor \(C_1+\cdots+C_n\). The main theorem of this paper is that if \(f\) is semistable and if \(g\geq 2\), then NEWLINE\[NEWLINE(\omega_{X/B}+ D)^2\leq (2g-2+n)(2q- 2+s)NEWLINE\]NEWLINE with equality if and only if \(f\) is isotrivial. It is proved by comparing the complete Poincaré metric \(\omega_P\) on \(B\backslash S\) (where \(S\) is the set of points lying below singular fibers) with the Weil-Petersson metric \(\omega_{WP}\) on the moduli space \({\mathcal M}_{g,n}\) of smooth projective curves of genus \(g\) with \(n\) distinct points removed.NEWLINENEWLINENEWLINEThe paper also claims to prove a height inequality for algebraic points on a curve defined over a function field (Theorem 0.2), but unfortunately the proof is not correct due to errors in the first paragraph of Section 3. This affects Corollary 0.3 as well.