On the Arakelov inequality in positive characteristic
Let \(k\) be an algebraically closed field of characteristic \(p\geq 3\). Let \(f: X\to C\) be a non-isotrivial semistable family of curves of genus \(g\geq 2\) defined over \(k\). Let \(b\) be the genus of the base curve \(C\), and let \(s\) be the number of singular fibers of \(f\). This paper provides the Arakelov inequality in positive characteristic, which is similar to the result of \textit{E. Viehweg} and \textit{K. Zuo} [J. Differ. Geom. 77, No. 2, 291--352 (2007; Zbl 1133.14010)] in characteristic zero. The result is obtained assuming that \(f\) is liftable to the ring \(W_2(k)\) of the second Witt vectors of \(k\). Theorem 1: Let \(k\) be an algebraically closed field of characteristic \(\text{char}(k)=p\geq 3\). Let \(f:X\to C\) be a non-isotrivial semistable family of curves of genus \(g\geq 2\) defined over \(k\). Assume that \(f\) is liftable to \(W_2(k)\). Then \[ \text{deg} f_*\omega_{X/C}\leq \frac{g-g_0}{2}(2b-2+s) \] where \(g_0\) is the rank of the kernel of the Higgs field associated to \(f\). Proof of Theorem 1 is similar to that of Viehweg and Zuo in characteristic zero case, but with crucial use of the semistablity of Higgs bundles corresponding to families of curves to estimate the degrees of the kernel and the image of the Higgs field. Theorem 2: Let \(f:X\to C\) be a non-isotrivial semistable family of curves of genus \(g\geq 2\) defined over \(k\). Assume that \(f\) is liftable to the ring \(W(k)\) of Witt vectors of \(k\). Then \[ \text{deg} f_*\omega_{X/C}\leq \frac{g}{2}(2b-2+s_1) \] where \(s_1\) is the number of singular fibers with non-compact Jacobians. Proof of Theorem 2 relies on the consistency of the relative invariants of \(f\) and its lifting. As a byproduct, an analogue of Beauville's conjecture in positive characteristic is presented when \(f\) is liftable to \(W(k)\). Corollary: Let \(f: X\to C={\mathbb{P}}^1\) as in Theorem 2. Then \(s\geq 5\). That is, there are at least \(5\) singular fibers for non-isotrivial semistable families of curves of genus \(g\geq 2\) over \({\mathbb{P}}^1\), which are liftable to \(W(k)\). The definition of a lifting of a non-isotrivial semistable family of curves of genus \(g\geq 2\) over \(k\) is the standard one. A family of curves \(f:X\to C\) with singular fibers \(F_1,\dots F_s\) is said to be liftable to \(W_2(k)\) if there is a semistable family \(\tilde{f}: \mathcal{X}\to C\) over \(W_2(k)\) and a semistable curve \(\sum_{i=1}^s\tilde{F}_i\) on \(\mathcal{X}\) such that the reduction of \(\tilde{f}\) to \(k\) is \(f\) and \(\sum_{i=1}^s\tilde{F}_i\) is the lifting of \(\sum_{i=1}^s F_i\) to \(W_2(k)\). A similar definition for the liftablity of \(f\) to \(W(k)\).
scientific article; zbMATH DE number 6898011
scientific article