The Collapsing Rate of the Kähler-Ricci Flow with Regular Infinite Time Singularity release_5dr7tsajzrf33ngqoloj4ipvfa

Abstract

We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^-t as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter e^-t/2. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with possibly singular fibers will also be discussed in local sense.
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