The collapsing rate of the Kähler–Ricci flow with regular infinite time singularity

Abstract

Abstract We study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>π</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Σ</m:mi> </m:mrow> </m:math> $\pi :X\rightarrow \Sigma $ inherited from its canonical bundle, where Σ is a Kähler manifold with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo form="prefix">dim</m:mo> <m:mi>ℂ</m:mi> </m:msub> <m:mi>Σ</m:mi> <m:mo>&lt;</m:mo> <m:msub> <m:mo form="prefix">dim</m:mo> <m:mi>ℂ</m:mi> </m:msub> <m:mi>X</m:mi> </m:mrow> </m:math> $\dim _\mathbb {C}\Sigma &amp;lt;\dim _\mathbb {C}X$ . We show that the flow metric degenerates at exactly the rate of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mi>t</m:mi> </m:mrow> </m:msup> </m:math> $e^{-t}$ as predicted by the cohomology information, and so the fibres <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>π</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $\pi ^{-1}(z)$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:mi>Σ</m:mi> </m:mrow> </m:math> $z \in \Sigma $ collapse at the optimal rate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo form="prefix">diam</m:mo> <m:mi>t</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>π</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>≃</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mi>t</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:math> $\operatorname{diam}_t (\pi ^{-1}(z)) \simeq e^{-t/2}$ . 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.