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Kähler–Ricci flow Type I conjecture

Let XX be a compact Kähler manifold, and let ω(t)\omega(t) solve the unnormalized Kähler–Ricci flow ∂tω(t)=−Ric⁡(ω(t))\partial_t\omega(t)=-\operatorname{Ric}(\omega(t)) on [0,T)[0,T), where T<∞T<\infty is the maximal time and αT=[ω(0)]−Tc1(X)\alpha_T=[\omega(0)]-T c_1(X) is the limiting cohomological class. For every fibre FF contracted by the contraction determined by αT\alpha_T, there should exist a constant CF>0C_F>0 such that, as t↗Tt\nearrow T, sup⁡x∈F∣R(ω(t))(x)∣≤CF/(T−t)\sup_{x\in F}|R(\omega(t))(x)|\leq C_F/(T-t) and diam⁡ω(t)(F)≤CFT−t\operatorname{diam}_{\omega(t)}(F)\leq C_F\sqrt{T-t}.

References

Primary source

arXiv

Additional references

geometric-analysiskhler-geometryricci-flow
Posted · edited

Progress summary

Refreshed
Claimed progress

The conjecture remains open in general, but new papers claim substantial Type I results for important collapsing Fano-bundle cases.

The conjecture predicts Type I control of finite-time Kähler–Ricci flow singularities, together with corresponding fibre-diameter control, across contracted fibres. No proposer or original date is supplied in the retrieved material.

Known results

  • Fong–Tran established partial results: local scalar-curvature bounds near Ricci vertices, almost-everywhere fibre-diameter bounds in collapsing Fano-fibre-bundle solutions, and the conjecture under symmetry assumptions.
  • In the blow-up case π:Bl⁡pY→Y\pi:\operatorname{Bl}_pY\to Y, Type-I parabolic limits along the exceptional divisor are claimed to be the Feldman–Ilmanen–Knopf shrinker Tot⁡(OPn−1(−1))\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1)).

September 2026 Fano-bundle progress

Jian and Song’s September 2026 work claims Type I control for collapsing Fano bundles; for one-dimensional fibres it claims a full-curvature bound, fibre diameters comparable to T−t\sqrt{T-t}, and tangent flows Cm×P1\mathbb{C}^m\times\mathbb{P}^1. Xiao’s September 9 revision independently claims Type I singularities for a Calabi-symmetric class of CPm\mathbb{C}P^m-bundles. These are substantial special-case advances, not a universal proof, and remain unverified.

Current status (as of September 2026): Type I behavior is claimed for several Fano-bundle and related special cases, while the universal conjecture for every contracted fibre remains open and unverified.

Sources

Solutions 0

No solutions have been posted yet.