A solution to Fujita's freeness conjecture via an extension theorem with analytic adjoint ideal sheaves

Abstract

The effective freeness in Fujita's conjecture states that, for an ample line bundle $L$ on a complex projective manifold $X$, the adjoint bundle $K_X\otimes L^{\otimes m}$ is globally generated when $m\ge {\dim }_{\mathbb{C}}X+1$. Following the approach of Angehrn and Siu, a solution is provided in this paper via the use of adjoint ideal sheaves, which provide a finer control of the non-integrable loci given by multiplier ideal sheaves, so that one can work directly with the lc singularities and the associated (minimal) lc centres as in the algebraic approaches of Kawamata and Helmke. The substitute for the Nadel or Kawamata-Viehweg vanishing theorem used in previous approaches is an extension theorem based on the techniques developed for the injectivity theorems.

Thanks to Prof. S. Helmke for pointing out that the construction of $f^{\text{loc}}$ on p.31 need not yield a local hol. section with the desired high vanishing order. Example: when $\mathtt{Z}$ locally behaves like $\{xy=z^2\}$, on which $x=\frac{z^2}{y}$ has vanishing order $2$ along $\{x=z=0\}$. It cannot be hol. extended to the ambient space locally with the vanishing order preserved