Abstract
We prove Bogomolov’s inequality for Higgs sheaves on varieties in positive characteristic $$p$$ that can be lifted modulo $$p^2$$ . This implies the Miyaoka–Yau inequality on surfaces of non-negative Kodaira dimension liftable modulo $$p^2$$ . This result is a strong version of Shepherd-Barron’s conjecture. Our inequality also gives the first algebraic proof of Bogomolov’s inequality for Higgs sheaves in characteristic zero, solving the problem posed by Narasimhan.
Highlights
Let X be a smooth projective variety of dimension n ≥ 2 defined over an algebraically closed field k
In [21] we proved that in positive characteristic the same inequality holds for strongly H -semistable sheaves
The first algebraic approach to Bogomolov’s inequality via bounding the number of sections of symmetric powers does not seem to work in the Higgs case. This motivated the author to work on the positive characteristic case and resulted in a proof of Bogomolov’s inequality for strongly semistable sheaves
Summary
Let X be a smooth projective variety of dimension n ≥ 2 defined over an algebraically closed field k. For any slope H -semistable sheaf E of rank r ≤ p = char k we have (E)H n−2 ≥ 0 This theorem was known only in the surface case for rank 2 vector bundles in characteristic 2 (see [34, Corollary 11]). The first algebraic approach to Bogomolov’s inequality via bounding the number of sections of symmetric powers does not seem to work in the Higgs case This motivated the author to work on the positive characteristic case and resulted in a proof of Bogomolov’s inequality for strongly semistable sheaves (see [21]). This did not shed light on the original problem.
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