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Existence of flips for generalized lc pairs
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abstract
We prove the existence of flips for $\mathbb Q$-factorial NQC generalized lc pairs, and the cone and contraction theorems for NQC generalized lc pairs. This answers a question of C. Birkar which was conjectured by J. Han and Z. Li. As an immediate application, we show that we can run the minimal model program for $\mathbb Q$-factorial NQC generalized lc pairs. In particular, we complete the minimal model program for $\mathbb Q$-factorial NQC generalized lc pairs in dimension $\leq 3$ and pseudo-effective $\mathbb Q$-factorial NQC generalized lc pairs in dimension $4$.
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Discreteness of volumes of divisors on Calabi-Yau type varieties
Volumes of integral divisors on epsilon-lc Calabi-Yau pairs lie in a fixed discrete set, settling Birkar's boundedness conjecture for polarized log Calabi-Yau pairs.
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