Let Y be a compact Kähler manifold, let p∈ Y, and let pi:M=operatornameBl_pY→ Y be the blow-up at p. Suppose that a Kähler–Ricci flow on M with initial Kähler class [omega_0] becomes singular at time T<∞ and satisfies [omega_0]-T c_1(M)=pi^[omega_Y]. Then every Type I parabolic blow-up limit of the flow along the exceptional divisor is the Feldman–Ilmanen–Knopf shrinking soliton on operatornameTot(O_P^n-1(-1)).

Feldman–Ilmanen–Knopf conjecture

The conjecture identifies the unique singularity model arising from a fundamental class of Kähler–Ricci-flow degenerations. Feldman, Ilmanen, and Knopf formulated it after proving uniqueness in the rotationally symmetric setting.

Known results

Feldman–Ilmanen–Knopf: uniqueness of the $U(n)$ -invariant complete shrinking soliton.
Maximo: a partial result in complex dimension $2$ under additional hypotheses.
2015: the conjecture in the $U(n)$ -invariant, non-collapsed flow on $\mathbb{CP}^{n}$ blown up at one point; collapsed flows instead limit to a compact soliton.
Cifarelli–Conlon–Deruelle: a strong compact-surface result, including bubble classification.

August 2026 preprint claim

A preprint claims that, for the stated compact Kähler blow-ups and finite-time $\mathrm{Type}\ I$ singularities, every exceptional-divisor parabolic blow-up converges to the unique FIK shrinker on $\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1))$ . The claim is unverified. The authors disclose that Chat GPT and Deepseek assisted with technical checking and exposition, not that they independently produced the proof.

Current status (as of August 2026): Special cases are established, while the new general compact-Kähler theorem remains an unverified preprint claim, so the conjecture is not yet settled.

Feldman–Ilmanen–Knopf conjecture

Primary source

Additional references

Progress summary

Solutions 0