A linear bound for Fujita's freeness conjecture

Abstract

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 $m\ge \lceil C_{0}n\rceil$, where $C_0=1.77629\dots$ 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 $|m{K}_Z|$ defines a birational map for every integer $m\ge 1+\lceil C_{0}n+3\sqrt{2\mathrm{e}n}\rceil$. Our main input is a new estimate for the multiplicity of a minimal log canonical center. If $(X,\Delta )$ 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 $2{\overline{e}}_1({\mathfrak{m}}_{W,x})\le (\dim W-{\mathrm{lct}}_x((X,\Delta );{\mathfrak{m}}_x)){\mathrm{mult}}_{x}W$, 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 ${\mathrm{mult}}_{x}W\le \frac{(a+c{)}^{a+c}}{a^a{c}^c}$, where $a:=\frac{\dim W-{\mathrm{lct}}_x((X,\Delta );{\mathfrak{m}}_x)}{2}$ and $c:=\mathrm{edim}{\mathcal{O}}_{W,x}-\dim W$.

25 pages; added effective birationality (Theorem 1.3), multiplicity of a klt singularity (Corollary 1.5), and two questions (Questions 5.10, 5.11)