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Diameter estimates in K\"ahler geometry
Diameter estimates for K\"ahler metrics are established which require only an
entropy bound and no lower bound on the Ricci curvature. The proof builds on
recent PDE techniques for estimates for the Monge-Amp\`ere equation,
with a key improvement allowing degeneracies of the volume form of codimension
strictly greater than one. As a consequence, diameter bounds are obtained for
long-time solutions of the K\"ahler-Ricci flow and finite-time solutions when
the limiting class is big, as well as for special fibrations of Calabi-Yau
manifolds
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