发表机构
Shanghai Center for Mathematical Sciences & School of Mathematical Sciences, Fudan University（上海数学中心与复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对光滑复射影簇的藤田自由性猜想，通过给出极小对数典范中心重数的新估计，证明了$K_X+mL$全局生成的线性界，确定了显式常数$C_0$及$K_X+2nL$的全局生成性。
AI 中文摘要
设$X$为$n$维光滑复射影簇，$L$为丰富卡蒂埃除子。我们证明对所有整数$m\geq\lceil C_0n\rceil$，$K_X+mL$是全局生成的，其中$C_0=1.77629\ldots$为显式常数，特别地$K_X+2nL$全局生成。核心贡献是对极小对数典范中心的重数给出新估计：若$(X,\Delta)$在闭点$x$附近对数典范但在$x$处非klt，$W$是过$x$的正维极小对数典范中心，则$2\overline{e}_1(\mathfrak m_{W,x})\leq\bigl(\dim W-\operatorname{lct}_x((X,\Delta);\mathfrak m_x)\bigr)\operatorname{mult}_xW$，其中$\overline{e}_1(\mathfrak m_{W,x})$是$\mathcal O_{W,x}$极大理想的第一正规希尔伯特系数，这推出$\operatorname{mult}_xW\leq \frac{(a+c)^{a+c}}{a^ac^c}$，其中$a:=\frac{\dim W-\operatorname{lct}_x((X,\Delta);\mathfrak m_{x})}{2}$，$c:=\operatorname{edim}\mathcal O_{W,x}-\dim W$。
英文摘要
Let $X$ be a smooth complex projective variety of dimension $n$, and let $L$ be an ample Cartier divisor. We prove that $K_X+mL$ is globally generated for every integer $m\geq\lceil C_0n\rceil$, where $C_0=1.77629\ldots$ is an explicit constant. In particular, $K_X+2nL$ is globally generated. We also prove that if $Z$ is a normal projective variety of dimension $n$ with $K_Z$ Cartier, nef and big, then $|mK_Z|$ defines a birational map for every integer $m\geq1+\lceil C_0n+3\sqrt{2\mathrm{e}n}\rceil$. Our main input is a new estimate for the multiplicity of a minimal log canonical center. If $(X,Δ)$ is log canonical near a closed point $x$ but is not klt at $x$, and $W$ is the positive-dimensional minimal log canonical center through $x$, then $2\overline{e}_1(\mathfrak m_{W,x})\leq\bigl(\dim W-\operatorname{lct}_x((X,Δ);\mathfrak m_x)\bigr)\operatorname{mult}_xW$, where $\overline{e}_1(\mathfrak m_{W,x})$ is the first normal Hilbert coefficient of the maximal ideal of $\mathcal O_{W,x}$. This implies $\operatorname{mult}_xW\leq \frac{(a+c)^{a+c}}{a^ac^c}$, where $a:=\frac{\dim W-\operatorname{lct}_x((X,Δ);\mathfrak m_{x})}{2}$ and $c:=\operatorname{edim}\mathcal O_{W,x}-\dim W$.
Comments25 pages; added effective birationality (Theorem 1.3), multiplicity of a klt singularity (Corollary 1.5), and two questions (Questions 5.10, 5.11)
