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

We study the collapsing behavior of the Kähler-Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion π: X → Σ inherited from its canonical bundle, where Σ is a Kähler manifold with dimℂ Σ < dimℂ X. We show that the flow metric degenerates at exactly the rate of e-t as predicted by the cohomology information, and so the fibres π-1 (z), z ∈ Σ collapse at the optimal rate diamt (π-1 (z)) ≃ e-t/2. Consequently, it leads to some analytic and geometric extensions to the regular case of works by J. Song and G. Tian. Its applicability to general Calabi-Yau fibrations will also be discussed in local settings. © 2015 by De Gruyter