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Orbifold stability and Miyaoka–Yau inequality for minimal pairs

Henri Guenancia and Behrouz Taji

Geometry & Topology 26 (2022) 1435–1482

After establishing suitable notions of stability and Chern classes for singular pairs, we use Kähler–Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log canonical pairs of log general type. We then proceed to prove the Miyaoka–Yau inequality for all minimal pairs with standard coefficients. Our result in particular provides an alternative proof of the abundance theorem for threefolds, which is independent of positivity results for cotangent sheaves established by Miyaoka.

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Miayoka-Yau inequality, minimal models, orbifold pairs, singular Kähler-Einstein metrics
Mathematical Subject Classification 2010
Primary: 14E20, 14E30, 32Q20
Secondary: 14C15, 14C17, 32Q26, 53C07
Received: 31 May 2017
Revised: 17 December 2020
Accepted: 17 April 2021
Published: 28 October 2022
Proposed: Lothar Göttsche
Seconded: Mark Gross, Dan Abramovich
Henri Guenancia
Institut de Mathématiques de Toulouse
Université de Toulouse
Behrouz Taji
School of Mathematics and Statistics
The University of Sydney
Sydney NSW