For pseudoeffective projective pairs, termination of one flip sequence implies termination of all, conditional on a conjecture about Nakayama-Zariski decomposition in the MMP.
Hu, Log abundance of the moduli b-divisors of lc-trivial fibrations, arXiv:2003.14379 0= 2020
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The normalization of the moduli space of polarized klt good minimal models of arbitrary Kodaira dimension is quasi-projective.
Proves generalized canonical bundle formula for lc-trivial fibrations without nef part assumption in complex analytic setting, plus algebraic counterpart.
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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.
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Quasi-Projective Moduli for Polarized klt Good Minimal Models
The normalization of the moduli space of polarized klt good minimal models of arbitrary Kodaira dimension is quasi-projective.
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Addendum: On generalized canonical bundle formula and boundedness of complements in complex analytic setting
Proves generalized canonical bundle formula for lc-trivial fibrations without nef part assumption in complex analytic setting, plus algebraic counterpart.