A Miyaoka-Yau Type Inequality for Threefolds with Nef Anti-canonical Divisors in Positive Characteristic

Abstract

We prove a Miyaoka–Yau type inequality for threefolds such that −K_X is nef and of numerical dimension ≥ 2

A\cdot (c_2(X)+\lambda c_1^2(X))>0,

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A \cdot \left(c_{2}(X)+\lambda{c}_{1}^{2}(X)\right) >0,$$\end{document}

where A is an ample line bundle and λ > 0 is a constant dependent on the characteristic p and the Cartier index n₀ of K_X. This is an analogue of the Miyaoka–Yau type inequality presented in [Algebra Number Theory, 2022, 16(10): 2339–2384], which treats minimal threefolds of general type. We attain this inequality by following the strategy of [Algebra Number Theory, 2022, 16(10): 2339–2384, Duke Math. J., 2019, 168(7): 1269–1301]. Applying a similar argument, we also show that in characteristic ≥ 5, if −K_X is nef and −K_X · c₂(X) < 0, then the nonvanishing theorem for anti-canonical divisor holds.

Keywords

Miyaoka–Yau inequality / threefold / positive characteristic / 14C17 / 14E99 / 14F99

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