## Duke Mathematical Journal

### The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic

#### Abstract

We generalize Bogomolov’s inequality for Higgs sheaves and the Bogomolov– Miyaoka–Yau inequality in positive characteristic to the logarithmic case. We also generalize Shepherd-Barron’s results on Bogomolov’s inequality on surfaces of special type from rank $2$ to the higher-rank case. We use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo $p^{2}$. These examples contradict some claims by Xie.

#### Article information

Source
Duke Math. J. Volume 165, Number 14 (2016), 2737-2769.

Dates
Revised: 12 October 2015
First available in Project Euclid: 2 June 2016

https://projecteuclid.org/euclid.dmj/1464872424

Digital Object Identifier
doi:10.1215/00127094-3627203

Mathematical Reviews number (MathSciNet)
MR3551772

Zentralblatt MATH identifier
06653511

#### Citation

Langer, Adrian. The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic. Duke Math. J. 165 (2016), no. 14, 2737--2769. doi:10.1215/00127094-3627203. https://projecteuclid.org/euclid.dmj/1464872424.

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