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

Let $\mathcal{F}$ be a canonical foliation of general type on a smooth projective surface $X$, and write $\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{F}):=\mathrm{v}\mathrm{o}\mathrm{l}(K_{\mathcal{F}})$. Let $G\subseteq \mathrm{Bir}(X,\mathcal{F})$ be a finite subgroup, and let $\mathcal{G}:=\mathcal{F}\mathrm{/}G$ be the quotient foliation in the birational sense. For a canonical foliation $\mathcal{H}$, define its $r$-th pluricanonical index by $${\delta }_r(\mathcal{H}):=\min \{m\in {\mathbb{Z}}_{>0}\mid h^0(m{K}_{\mathcal{H}})\ge r\},$$ where $\min \mathrm{\varnothing }:=\mathrm{\infty }$. For an arbitrary foliation, these indices are computed on any canonical birational model. If $\kappa (\mathcal{G})\ge 0$, we prove $$\mathrm{\mid }G\mathrm{\mid }\le \{\begin{array}{ll}4{\delta }_1(\mathcal{G})\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{F}),& \kappa (\mathcal{G})=0,\\ {\displaystyle \frac{4}{3}{\delta }_2(\mathcal{G})\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{F}),}& \kappa (\mathcal{G})=1,\\ {\delta }_2(\mathcal{G}{)}^2(1+{\delta }_2(\mathcal{G}))\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{F}),& \kappa (\mathcal{G})=2.\end{array}$$ Since $\mathrm{Bir}(X,\mathcal{F})$ is finite, one may in particular take $G=\mathrm{Bir}(X,\mathcal{F})$. When $\kappa (\mathcal{G})=0$ or $1$, the effective bounds ${\delta }_1(\mathcal{G})\le 12$ and ${\delta }_2(\mathcal{G})\le 42$ give $$\mathrm{\mid }G\mathrm{\mid }\le 48\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{F})\text{and}\mathrm{\mid }G\mathrm{\mid }\le 56\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{F}),$$ 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.