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.
On the existence of minimal models for log canonical pairs
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We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.
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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.