The Orbibundle Miyaoka-Yau-Sakai Inequality and an Effective Bogomolov-McQuillanTheorem

Abstract

Let $X$ be a minimal projective surface of general type defined over the complex numbers and let $C \subset X$ be an irreducible curve of geometric genus $g$. Given a rational number $\alpha \in [0,1]$, we construct an orbibundle $\tilde{\mathcal{E}}_{\alpha}$ associated with the pair $(X,C)$ and establish the Miyaoka-Yau-Sakai inequality for $\tilde{\mathcal{E}}_{\alpha}$. By varying the parameter $\alpha$ in the inequality, we derive several geometric consequences involving the ``canonical degree'' $CK_X$ of $C$. Specifically we prove the following two results. (1) If $K_X^2$ is greater than the topological Euler number $c_2(X)$, then $CK_X$ is uniformly bounded from above by a function of the invariants $g, K_X^2$ and $c_2(X)$ (an effective version of a theorem of Bogomolov-McQuillan). (2) If $C$ is nonsingular, then $CK_X \leq 3g - 3 + o(g)$ when $g$ is large compared to $K_X^2, c_2(X)$ (an affirmative answer to a conjecture of McQuillan).