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

Abstract

Abstract For any integer n > 0 n>0 , the 𝑛-th canonical stability index r n r_{n} is defined to be the smallest positive integer so that the r n r_{n} -canonical map Φ r n \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 } \{r_{n}\} as follows: r n r_{n} is equal to the maximum of the set of those canonical stability indices of smooth projective ( n + 1 ) (n+1) -folds with sufficiently large canonical volumes. Equivalently, there exists a constant V ⁢ ( n ) > 0 \mathfrak{V}(n)>0 such that, for any smooth projective 𝑛-fold 𝑋 with the canonical volume vol ⁢ ( X ) > V ⁢ ( n ) \mathrm{vol}(X)>{\mathfrak{V}}(n) , the pluricanonical map φ m , X \varphi_{m,X} is birational onto the image for all m ≥ r n − 1 m\geq r_{n-1} .