JournalsprimsVol. 44, No. 2pp. 403–417

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

  • Yoichi Miyaoka

    University of Tokyo, Japan
The Orbibundle Miyaoka–Yau–Sakai Inequality and an Effective Bogomolov–McQuillan Theorem cover
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Abstract

Let X be a minimal projective surface of general type defined over the complex numbers and let CX 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