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 $C\subset X$ 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” $C{K}_X$ 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 $C{K}_X$ 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 $C{K}_X\le 3g-3+o(g)$ when $g$ is large compared to $K_X^2$, $c_2(X)$ (an affrmative answer to a conjecture of McQuillan).