Birational Automorphism Bounds for General-Type Foliations on Surfaces via Pluricanonical Indices

Abstract

Let $\mathcal{F}$ be a canonical foliation of general type on a smooth projective surface $X$, and write $\mathrm{vol}(\mathcal{F}):=\mathrm{vol}(K_{\mathcal{F}})$. Let $G\subseteq\operatorname{Bir}(X,\mathcal{F})$ be a finite subgroup, and let $\mathcal{G}:=\mathcal{F}/G$ be the quotient foliation in the birational sense. For a canonical foliation $\mathcal{H}$, define its $r$-th pluricanonical index by \[ \delta_r(\mathcal{H}) := \min \bigl\{ m\in\mathbb{Z}_{>0} \mid h^0(mK_{\mathcal{H}})\geq r \bigr\}, \] where $\min\varnothing:=\infty$. For an arbitrary foliation, these indices are computed on any canonical birational model. If $\kappa(\mathcal{G})\geq0$, we prove \[ |G| \leq \begin{cases} 4\delta_1(\mathcal{G})\,\mathrm{vol}(\mathcal{F}), &\kappa(\mathcal{G})=0,\\[1mm] \displaystyle \frac{4}{3}\delta_2(\mathcal{G})\,\mathrm{vol}(\mathcal{F}), &\kappa(\mathcal{G})=1,\\[3mm] \delta_2(\mathcal{G})^2 \bigl(1+\delta_2(\mathcal{G})\bigr)\, \mathrm{vol}(\mathcal{F}), &\kappa(\mathcal{G})=2. \end{cases} \] Since $\operatorname{Bir}(X,\mathcal{F})$ is finite, one may in particular take $G=\operatorname{Bir}(X,\mathcal{F})$. When $\kappa(\mathcal{G})=0$ or $1$, the effective bounds $\delta_1(\mathcal{G})\leq12$ and $\delta_2(\mathcal{G})\leq42$ give \[ |G|\leq48\,\mathrm{vol}(\mathcal{F}) \qquad\text{and}\qquad |G|\leq56\,\mathrm{vol}(\mathcal{F}), \] respectively. The main new ingredient is a cluster formula for adjoint volumes, which yields index-dependent lower bounds for tangency-free foliated surface pairs whose underlying foliation has Kodaira dimension zero or one.