# The Orbibundle Miyaoka–Yau–Sakai Inequality and an Eﬀective Bogomolov–McQuillan Theorem

### Yoichi Miyaoka

University of Tokyo, Japan

## 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 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. 403–417

DOI 10.2977/PRIMS/1210167331