Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded, which implies birational boundedness for stable families of maximal variation.
Fan, Volumes of foliations birationally bounded by algebraically integrable families
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Birational boundedness of stable families
Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded, which implies birational boundedness for stable families of maximal variation.