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

Abstract For any integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> n&gt;0 , the 𝑛-th canonical stability index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>r</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> r_{n} is defined to be the smallest positive integer so that the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>r</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> r_{n} -canonical map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Φ</m:mi> <m:msub> <m:mi>r</m:mi> <m:mi>n</m:mi> </m:msub> </m:msub> </m:math> \Phi_{r_{n}} is stably birational onto its image for all smooth projective 𝑛-folds of general type. We prove the lifting principle for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{r_{n}\} as follows: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>r</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> r_{n} is equal to the maximum of the set of those canonical stability indices of smooth projective <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (n+1) -folds with sufficiently large canonical volumes. Equivalently, there exists a constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="fraktur">V</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> \mathfrak{V}(n)&gt;0 such that, for any smooth projective 𝑛-fold 𝑋 with the canonical volume <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>vol</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>&gt;</m:mo> <m:mrow> <m:mi mathvariant="fraktur">V</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \mathrm{vol}(X)&gt;{\mathfrak{V}}(n) , the pluricanonical map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>φ</m:mi> <m:mrow> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>X</m:mi> </m:mrow> </m:msub> </m:math> \varphi_{m,X} is birational onto the image for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:msub> <m:mi>r</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math>