Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

Abstract: We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

• The paper establishes a fixed-gauge, potential-level description of Kähler–Ricci flows near finite-time singularities, with quantitative derivative decay and uniqueness of the tangent shrinker under holomorphic approximation.
• It applies the stability theory to surface blowdowns, Calabi-ansatz flips, divisorial contractions, and orbifold contractions, identifying explicit shrinker and expander models such as the FIK shrinker and Cao expander.
• The results rely on drift-Laplacian spectral analysis, nonlinear potential estimates, non-concentration, and pseudolocality, while leaving holomorphic approximation and the proposed Fano-fibration characterization open in general dimensions.

Overview

This paper, by Longteng Chen, Max Hallgren, and Lucas Lavoyer (2608.19152), establishes a fixed-gauge, potential-level description of Kähler–Ricci flows through finite-time singularities arising in the analytic minimal model program of Song and Tian. The central result (Theorem 1.1 of the paper) states that if a compact Kähler–Ricci flow developing a singularity at time $t=0$ can be holomorphically approximated by an asymptotically conical (AC) shrinking Kähler–Ricci soliton $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$, then on a neighborhood of the exceptional set there exist biholomorphisms $F$ and $\widehat{F}$ identifying the flow with its tangent shrinker, such that

$F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$

with quantitative decay estimates for the relative potential and all its derivatives:

$|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$

The logarithmic loss in this estimate is sharp for the method and reflects the slow convergence inherent to the parabolic problem. The theorem upgrades previously known tangent-flow convergence — which holds only along rescaling sequences with varying gauges — to convergence in a single fixed holomorphic gauge, and it forces uniqueness of the tangent flow. The authors state that these are the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

Background: Song's conjectural picture

The analytic minimal model program seeks metric counterparts of birational surgeries encoded by diagrams $M \to Y \leftarrow M'$, where $\pi$ and $\pi'$ contract holomorphic curves. Song and Tian constructed canonical Kähler–Ricci flows through singularities realizing these contractions in the sense of currents, and conjectured Gromov–Hausdorff continuity of the flow through the singular time. Song–Weinkove confirmed this continuity for complex surfaces, where the surgery is the blowdown of finitely many disjoint $(-1)$-curves.

The refined conjecture of Jian–Song predicts Type I curvature bounds approaching and emerging from the singular time, together with pointed Cheeger–Gromov convergence of parabolic dilations to self-similar flows generated by AC shrinking and expanding solitons sharing a common asymptotic cone $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$0 — the unique tangent cone of the singular space at the contracted point. In particular, the ancient shrinker flow and immortal expander flow should form an infinitesimal model $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$1 of the surgery.

Prior work established the incoming half of this picture (Type I bounds and tangent-flow convergence) in several settings. What was missing is a description valid in a fixed gauge on a definite spacetime region, and any control of the outgoing flow. This paper supplies both, conditional on holomorphic approximation.

Main results

The paper's contributions can be summarized as follows:

• Fixed-gauge stability near AC shrinkers (main theorem): under holomorphic approximation by an AC shrinker, the flow is a $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$2-exact perturbation of the shrinker flow in a fixed holomorphic gauge, with decay estimates as above; moreover the base $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$3 acquires isolated conical singularities modeled on the asymptotic cone, and the outgoing flow is likewise controlled.
• Complex dimension two (Corollary on surfaces): every noncollapsed Kähler–Ricci flow through a blowdown of $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$4-curves is modeled, in a fixed gauge, on the Feldman–Ilmanen–Knopf (FIK) shrinker on $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$5 converging to the cone metric $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$6 on $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$7, with the Cao expander modeling the emerging flow. This confirms a strong form of Song's conjecture in dimension two.
• Calabi ansatz flips and divisorial contractions: higher-dimensional analogues using the unique $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$8-invariant AC shrinkers and expanders on total spaces of $(M_{\sol},J_{\sol},g_{\sol},f_{\sol})$9 and $F$0, asymptotic to cones over $F$1.
• Orbifold contractions: a Calabi-ansatz flow contracting to a smooth orbifold point, with the outgoing model an orbifold expander on $F$2.
• Compact shrinker case (second main theorem): for any compact Kähler–Ricci shrinker and any initial metric in $F$3, the flow satisfies $F$4 with $F$5 — exponential convergence in modified time, without symmetry assumptions on the initial data. This improves prior rates of Deruelle–Schulze and Tian–Zhang–Zhang and answers affirmatively a question raised by Zhu.

A notable structural point: the authors formulate Conjecture 1.7, proposing that holomorphic approximation by a shrinker holds if and only if the Fano fibration germ $F$6 agrees with the polarized Fano fibration of some Kähler–Ricci shrinker, based on ideas of Sun–Zhang. This remains open outside the cases covered here.

Method of proof

The proof converts the problem into a nonlinear stability question for the modified Kähler–Ricci flow

$F$7

where $F$8 is the soliton vector field and $F$9 is the normalized time variable. Writing $\widehat{F}$0 relative to a family of model metrics $\widehat{F}$1, the potential satisfies a complex Monge–Ampère type equation whose linearization at zero is the drift heat equation

$\widehat{F}$2

Several technical pillars support the argument:

• Spectral analysis of the drift Laplacian. Using a weighted Weitzenböck formula and a closed-range argument for $\widehat{F}$3 on $\widehat{F}$4, the authors show that eigenvalues below $\widehat{F}$5 correspond precisely to pluriharmonic functions of sub-quadratic growth, while the $\widehat{F}$6 eigenspace is spanned by pluriharmonic functions and the finite-dimensional space $\widehat{F}$7 of holomorphy potentials for real-holomorphic Killing fields commuting with $\widehat{F}$8. These neutral and unstable directions correspond to changes of gauge rather than genuine metric instabilities. Sharp pointwise growth estimates $\widehat{F}$9 for eigenfunctions are established via weighted $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$0 averaging inequalities in the spirit of Colding–Minicozzi.
• Model metrics. Following Chiu–Szekelyhidi, the family $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$1, obtained from biholomorphisms generated by elements of $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$2, absorbs the non-pluriharmonic neutral modes. Uniform comparability estimates for potentials, tensors, and sublevel sets $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$3 are proved.
• Non-concentration. A key lemma shows that smallness of $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$4 in weighted $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$5 on a large sublevel set implies an $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$6 bound ($F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$7) on a larger sublevel set one unit of modified time later. The positive part is handled by grafting $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$8 onto a normalization of $F^{\ast}\widetilde{\omega}_t = \omega_{\sol,t}+\sqrt{-1}\partial \overline{\partial} \widetilde{\varphi}_t,$9 to obtain a global subsolution of the drift heat equation, then applying hypercontractivity; the negative part requires grafting onto a global heat solution comparable to the soliton potential, using precise conjugate heat kernel estimates of Fang–Li together with pseudolocality and center estimates from Bamler's compactness theory.
• Decay proposition and iteration. By contradiction-compactness, any solution small in a large region admits a gauge modification (change of model parameter $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$0 plus subtraction of unstable pluriharmonic modes) after which the weighted $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$1 functional decays by a factor $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$2 per unit time, where $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$3 is determined by the first stable eigenvalue. Iterating yields a convergent sequence $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$4 and exponentially decaying potentials on regions of size $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$5. Translating back to un-normalized time gives fast convergence on regions of size approximately $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$6 as $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$7; a final pseudolocality argument propagates the estimates to the full region $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$8 up to $|(\nabla^{g_{\sol,t}})^k\widetilde{\varphi}_t|_{g_{\sol,t}} \leq C_k \left( r(x)+\sqrt{|t|} \right)^{2-k} \left| \log \left( r(x)+\sqrt{|t|}\right) \right|^{-1}.$9, producing the biholomorphisms $M \to Y \leftarrow M'$0 and $M \to Y \leftarrow M'$1 and the derivative estimates.

An important consequence established en route: the limiting shrinker and holomorphic gauge are independent of the rescaling sequence, giving uniqueness of the tangent flow.

Applications to specific geometries

In complex dimension two, the classification input is strong: by Cifarelli–Conlon–Deruelle, every nontrivial tangent flow of a noncollapsed surface flow is holomorphically isometric to the FIK shrinker, and the cohomological vanishing $M \to Y \leftarrow M'$2 (proved via the affine bundle structure and relative GAGA) guarantees holomorphic approximation. Consequently each contracted $M \to Y \leftarrow M'$3-curve admits a fixed-gauge embedding of the FIK region, the base acquires isolated cone points modeled on $M \to Y \leftarrow M'$4, and the outgoing orbifold-free flow is modeled on the Cao expander with matching boundary potentials $M \to Y \leftarrow M'$5.

For the Calabi ansatz flip, the uniqueness results of Li for invariant shrinkers and expanders supply both models, and Jian–Song provide the Type I bound and tangent-flow identification; the paper then upgrades these to the fixed-gauge statement with commuting diagrams of embeddings $M \to Y \leftarrow M'$6, $M \to Y \leftarrow M'$7, $M \to Y \leftarrow M'$8.

The orbifold case requires a minor extension of the Chen–Hallgren–Lee gluing construction to expanders that are global $M \to Y \leftarrow M'$9-quotients of smooth ones; the authors verify that the maximum principle and pseudolocality arguments carry over to smooth orbifolds, and identify the glued limit flow with the Song–Weinkove continuation via log terminality of the orbifold singularities.

Limitations and open questions

The main theorem is conditional on the holomorphic approximation hypothesis (Definition 1.2), i.e., existence of approximating embeddings satisfying $\pi$0 exactly. While Lemma 2.14 provides a sufficient criterion — vanishing of $\pi$1 for an exhaustion by strictly pseudoconvex sets — and this criterion is verified for the FIK shrinker and implicitly available in the symmetric settings above, it is not known to hold for a general AC shrinker tangent to a general flow. The proposed characterization (Conjecture 1.7) in terms of Fano fibration germs remains open, including in higher-dimensional situations without symmetry. In higher dimensions without Calabi ansatz or large symmetry groups, even the Gromov–Hausdorff continuity of the flow through singularities — let alone the fixed-gauge description — remains conjectural. Finally, the decay exponent $\pi$2 depends on the spectral gap $\pi$3 of the drift Laplacian, so no explicit uniform rate is available without spectral information on the specific shrinker.

Conclusion

The paper delivers a complete infinitesimal description of Kähler–Ricci flow surgeries in three regimes: noncollapsed surface flows, Calabi-ansatz flips and divisorial contractions, and orbifold contractions, confirming a strong form of Song's conjecture in each. Technically, it combines spectral theory of the drift Laplacian on shrinkers, a gauge-flexible nonlinear stability scheme adapted from Chiu–Szekelyhidi, non-concentration estimates built on hypercontractivity and Bamler-style heat kernel bounds, and pseudolocality propagation. The compact-shrinker theorem additionally yields unconditional, symmetry-free exponential convergence of any Fano-class flow to its unique shrinker at the level of Kähler potentials. The principal open problem left by the paper is to verify the holomorphic approximation hypothesis, or equivalently Conjecture 1.7, beyond the symmetric and two-dimensional settings treated here.