Type-I fibre-diameter bound for Kähler–Ricci contractions

Prove the Type I fibre-diameter estimate \(\operatorname{diam}_{d_t}\pi^{-1}(q)\leq C\sqrt{T-t}\) for the fibres of a bimeromorphic contraction arising from a finite-time noncollapsing Kähler–Ricci flow.

Background

For a bimeromorphic contraction π:X→Y\pi:X\to Y, Proposition 8.4 identifies the Gromov–Hausdorff limit with the target precisely when the fibres collapse uniformly in diameter. Existing results establish weaker fibre-diameter decay, such as a C(T−t)1/3C(T-t)^{1/3} bound for certain contractions.

The paper identifies the stronger CT−tC\sqrt{T-t} estimate as the second half of a conjecture stated in Jian–Song–Tian. The authors explicitly note that their heat-kernel estimates do not establish this bound, because localization of a conjugate heat-kernel measure around its base point does not imply localization around an entire fibre.

References

The stronger Type I fibre diameter bound $$\diam_{d_t}\pi{-1}(q)\leq C\sqrt{T-t}$$ is the second half of Conjecture 1.1.

— Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits  (2610.21824 - Zhang et al., 15 Oct 2026) in Remark 8.5, “fibreremark”