A lifting principle for canonical stability indices of varieties of general type

Abstract

For any integer $n>0$, the $n$th canonical stability index $r_n$ is defined to be the smallest positive integer so that the $r_n$-canonical map ${\Phi }_{r_n}$ is stably birational onto its image for all smooth projective $n$-folds of general type. We prove the lifting principle for $\{r_n\}$ as follows: $r_n$ equals to the maximum of the set of those canonical stability indices of smooth projective $(n+1)$-folds with sufficiently large canonical volumes. Equivalently, there exists a constant $\mathfrak{V}(n)>0$ such that, for any smooth projective $n$-fold $X$ with the canonical volume $\text{vol}(X)>\mathfrak{V}(n)$, the pluricanonical map ${\phi }_{m,X}$ is birational onto the image for all $m\ge r_{n-1}$. The ''lifting principle'' was first put forward by James McKernan in Mathematics Review (MR2339333).

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