The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem

Abstract

Let X be a minimal projective surface of general type defined over the complex numbers and let C ⊆ X be an irreducible curve of geometric genus g. Given a rational number a ∈ [0, 1], we construct an orbibundle ε̄ α associated with the pair (X, C) and establish the Miyaoka-Yau-Sakai inequality for ε̄α. By varying the parameter α in the inequality, we derive several geometric consequences involving the "canonical degree" CKX of C. Specifically we prove the following two results. (1) If KX2 is greater than the topological Euler number c2(X), then CKX is uniformly bounded from above by a function of the invariants g, KX2 and c2(X) (an effective version of a theorem of Bogomolov-McQuillan). (2) If C is nonsingular, then CKX ≤ 3g - 3 + o(g) when g is large compared to KX2, c2(X) (an affirmative answer to a conjecture of McQuillan). © 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.