发表机构
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对光滑射影曲面上一般型典范叶状结构的双有理自同构群，利用多重典范指标与伴随体积的簇公式，推导得到不同 Kodaira 维数下的自同构群阶的上界。
AI 中文摘要
设$\boldsymbol{\textit{X}}$为光滑射影曲面，$\boldsymbol{\textit{F}}$为$\boldsymbol{\textit{X}}$上的一般型典范叶状结构，记$\boldsymbol{\textit{vol}}(\boldsymbol{\textit{F}}):=\boldsymbol{\textit{vol}}(\boldsymbol{\textit{K}}_{\boldsymbol{\textit{F}}})$。设$\boldsymbol{\textit{G}}\boldsymbol{\boldsymbol{\text{⊆}}}\boldsymbol{\textit{Bir}}(\boldsymbol{\textit{X}},\boldsymbol{\textit{F}})$为有限子群，$\boldsymbol{\textit{G}}$对应的双有理商叶状结构记为$\boldsymbol{\textit{G}}:=\boldsymbol{\textit{F}}/\boldsymbol{\textit{G}}$。对典范叶状结构$\boldsymbol{\textit{H}}$，定义其第$\boldsymbol{\textit{r}}$个多重典范指标为$\boldsymbol{\textit{δ}}_{\boldsymbol{\textit{r}}}(\boldsymbol{\textit{H}}):= \boldsymbol{\textit{min}} \bigl\{ \boldsymbol{\textit{m}}\boldsymbol{\boldsymbol{\text{∈}}}\boldsymbol{\textit{Z}}_{>0} \boldsymbol{\boldsymbol{\text{|}}} \boldsymbol{\textit{h}}^{0}(\boldsymbol{\textit{m}}\boldsymbol{\textit{K}}_{\boldsymbol{\textit{H}}})\boldsymbol{\boldsymbol{\text{≥}}}\boldsymbol{\textit{r}} \bigr\}$，其中空集的最小值定义为无穷大；对任意叶状结构，这些指标可在任意典范双有理模型上计算。若$\boldsymbol{\textit{κ}}(\boldsymbol{\textit{G}})\boldsymbol{\boldsymbol{\text{≥}}}\boldsymbol{\text{0}}$，本文证明：当$\boldsymbol{\textit{κ}}(\boldsymbol{\textit{G}})\boldsymbol{\text{=}}\boldsymbol{\text{0}}$时，$\boldsymbol{|\boldsymbol{\textit{G}}|}\boldsymbol{\text{≤}}\boldsymbol{\text{4}}\boldsymbol{\textit{δ}}_{\boldsymbol{\text{1}}}(\boldsymbol{\textit{G}})\boldsymbol{\textit{vol}}(\boldsymbol{\textit{F}})$；当$\boldsymbol{\textit{κ}}(\boldsymbol{\textit{G}})\boldsymbol{\text{=}}\boldsymbol{\text{1}}$时，$\boldsymbol{|\boldsymbol{\textit{G}}|}\boldsymbol{\text{≤}}\boldsymbol{\text{(4/3)}}\boldsymbol{\textit{δ}}_{\boldsymbol{\text{2}}}(\boldsymbol{\textit{G}})\boldsymbol{\textit{vol}}(\boldsymbol{\textit{F}})$；当$\boldsymbol{\textit{κ}}(\boldsymbol{\textit{G}})\boldsymbol{\text{=}}\boldsymbol{\text{2}}$时，$\boldsymbol{|\boldsymbol{\textit{G}}|}\boldsymbol{\text{≤}}\boldsymbol{\textit{δ}}_{\boldsymbol{\text{2}}}(\boldsymbol{\textit{G}})^{\boldsymbol{\text{2}}}\boldsymbol{(}\boldsymbol{\text{1}}\boldsymbol{+}\boldsymbol{\textit{δ}}_{\boldsymbol{\text{2}}}(\boldsymbol{\textit{G}})\boldsymbol{)}\boldsymbol{\textit{vol}}(\boldsymbol{\textit{F}})$。由于$\boldsymbol{\textit{Bir}}(\boldsymbol{\textit{X}},\boldsymbol{\textit{F}})$是有限群，可取$\boldsymbol{\textit{G}}\boldsymbol{\text{=}}\boldsymbol{\textit{Bir}}(\boldsymbol{\textit{X}},\boldsymbol{\textit{F}})$；当$\boldsymbol{\textit{κ}}(\boldsymbol{\textit{G}})\boldsymbol{\text{=}}\boldsymbol{\text{0}}$或$\boldsymbol{\text{1}}$时，利用有效界$\boldsymbol{\textit{δ}}_{\boldsymbol{\text{1}}}(\boldsymbol{\textit{G}})\boldsymbol{\text{≤}}\boldsymbol{\text{12}}$、$\boldsymbol{\textit{δ}}_{\boldsymbol{\text{2}}}(\boldsymbol{\textit{G}})\boldsymbol{\text{≤}}\boldsymbol{\text{42}}$，可得$\boldsymbol{|\boldsymbol{\textit{G}}|}\boldsymbol{\text{≤}}\boldsymbol{\text{48}}\boldsymbol{\textit{vol}}(\boldsymbol{\textit{F}})$和$\boldsymbol{|\boldsymbol{\textit{G}}|}\boldsymbol{\text{≤}}\boldsymbol{\text{56}}\boldsymbol{\textit{vol}}(\boldsymbol{\textit{F}})$。本文的核心新要素是伴随体积的簇公式，该公式为底叶状结构的 Kodaira 维数为 0 或 1 的无切触叶状曲面对，给出了依赖指标的下界。
英文摘要
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 \[ δ_r(\mathcal{H}) := \min \bigl\{ m\in\mathbb{Z}_{>0} \mid h^0(mK_{\mathcal{H}})\geq r \bigr\}, \] where $\min\varnothing:=\infty$. For an arbitrary foliation, these indices are computed on any canonical birational model. If $κ(\mathcal{G})\geq0$, we prove \[ |G| \leq \begin{cases} 4δ_1(\mathcal{G})\,\mathrm{vol}(\mathcal{F}), &κ(\mathcal{G})=0,\\[1mm] \displaystyle \frac{4}{3}δ_2(\mathcal{G})\,\mathrm{vol}(\mathcal{F}), &κ(\mathcal{G})=1,\\[3mm] δ_2(\mathcal{G})^2 \bigl(1+δ_2(\mathcal{G})\bigr)\, \mathrm{vol}(\mathcal{F}), &κ(\mathcal{G})=2. \end{cases} \] Since $\operatorname{Bir}(X,\mathcal{F})$ is finite, one may in particular take $G=\operatorname{Bir}(X,\mathcal{F})$. When $κ(\mathcal{G})=0$ or $1$, the effective bounds $δ_1(\mathcal{G})\leq12$ and $δ_2(\mathcal{G})\leq42$ 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.
Comments38 pages
