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 α ∈ [0, 1], we construct an orbibundle _Ẽ_α associated with the pair (X, C) and establish the Miyaoka–Yau–Sakai inequality for _Ẽ_α. 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 K2X is greater than the topological Euler number c2(X), then CKX is uniformly bounded from above by a function of the invariants g, K2X 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 K2X, c2(X) (an affrmative answer to a conjecture of McQuillan).
Cite this article
Yoichi Miyaoka, The Orbibundle Miyaoka–Yau–Sakai Inequality and an Eﬀective Bogomolov–McQuillan Theorem. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, pp. 403–417DOI 10.2977/PRIMS/1210167331