On the Arakelov inequality in positive characteristic
type: Item
instance of: scholarly article
title: On the Arakelov inequality in positive characteristic
zbMATH Open document ID: 1393.14009
author: Wan-Yuan Xu
published in: Mathematische Zeitschrift
publication date: 2018-07-02T00:00:00Z
review text: 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 semistabl…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 6898011
zbMATH Keywords: family of curves
zbMATH Keywords: Arakelov inequality
zbMATH Keywords: Higgs bundle
zbMATH Keywords: Beauville's conjecture
MaRDI profile type: Publication
full work available at URL: s00209-017-1945-5
OpenAlex ID: W2765986094
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DOI: 10.1007/S00209-017-1945-5
description: scientific article; zbMATH DE number 6898011