Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded, which implies birational boundedness for stable families of maximal variation.
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For pseudoeffective projective pairs, termination of one flip sequence implies termination of all, conditional on a conjecture about Nakayama-Zariski decomposition in the MMP.
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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.
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Nakayama-Zariski decomposition and the termination of flips
For pseudoeffective projective pairs, termination of one flip sequence implies termination of all, conditional on a conjecture about Nakayama-Zariski decomposition in the MMP.