Neckpinch — Kähler–Ricci flow

Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

Sat, 17 Oct 2026 00:00:00 GMT

Randall R. Van Why — We demonstrate that Kähler-Ricci flow on a Fano surface decomposes as a symplectic deflation along a plurianticanonical divisor plus an exact symplectic deformation in the divisor complement. We apply this decomposition to the study of certain projective bundles. We establish the optimal decay rate in the setting of Song, Székelyhidi, and Weinkove for a family of toric metrics which match the cohomological profile of Kähler-Ricci flow. In doing so, we identify a C²-obstruction to proving the optimal decay rate in the setting of ruled toric surfaces. We present these ideas as a new symplectic-topological approach to the study of Kähler-Ricci flow.

Type I estimates for the Kähler-Ricci flow II

Fri, 16 Oct 2026 00:00:00 GMT

Wangjian Jian, Jian Song — In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial C⁰ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

Thu, 15 Oct 2026 00:00:00 GMT

Frederick Tsz-Ho Fong, Junbang Liu — Let X be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class [ω], Jian–Shi–Song proved that there exists a unique cscK metric in the class 2πc₁(K_X)+ε[ω] when ε∈ (0,ε₀], using the properness criterion of Chen–Cheng. We prove that the cscK metrics in 2πc₁(K_X)+ε[ω] converge, as εdownarrow0, in the Gromov–Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of X as conjectured by Jian–Shi–Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee–Tosatti–Zhang on the Gromov-Hausdorff convergence of the normalized Kähler–Ricci flow on X.

Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

Thu, 15 Oct 2026 00:00:00 GMT

Lei Zhang, Zhenlei Zhang — In this paper, we study heat kernel estimates along the Kähler-Ricci flow with finite time noncollapsing singularities over a compact Kähler manifold. The starting point is the type-II scalar-curvature estimate, which gives the natural parabolic scale r≲ T-t near the singular time. By combining the distance distortion and space-time cut-off estimates of Bamler–Zhang with entropy, mean-value inequality, and conjugate heat-kernel estimates, we obtain two-sided Gaussian bounds and gradient bounds that are uniform on this scale. Under the global type-I scalar-curvature estimate, the same estimates hold on every scale. We then use these estimates and the monotonicity of the W₁-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances dₜ (induced by ω(t)) converge uniformly to this pseudo-distance. Its metric quotient is the unique Gromov–Hausdorff limit. Finally, we apply Bamler's theory and the recent work of Hallgren–Zhang to obtain the following. Every fixed-base-point type-I blow-up admits a subsequential metric-flow limit whose time slices are singular shrinking Kähler–Ricci soliton spaces with normal complex-analytic and log-terminal structure, unique tangent cones, and a polarized Fano fibration. For moving base points, the limiting metric flow, equipped with its limiting conjugate heat flow, is a metric soliton precisely under an entropy-pinching condition.

Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

Sun, 04 Oct 2026 00:00:00 GMT

Liding Huang, Xiaohua Zhu — We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold (M,J,T), and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial T-invariant generalized Kähler metric with its symplectic form in 2πc₁(M,J), we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.

Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

Thu, 01 Oct 2026 00:00:00 GMT

Yeyun Xu, Linfeng Zhou — We establish a Type I curvature bound for finite-time volume-collapsing Kähler–Ricci flows on compact Kähler surfaces. No symmetry assumption is imposed, and the initial Kähler class need not be rational. The proof combines local symplectic topology with the classification of gradient Kähler–Ricci shrinking solitons. Bamler's compactness and structure theory provides the shrinking limits used in the argument.

Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

Thu, 01 Oct 2026 00:00:00 GMT

Jiyuan Han — We prove a uniform L^∞-estimate for the parabolic Monge–Ampère equation under uniform Hörmander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled Kähler–Ricci flows on a family of polarized Calabi–Yau manifolds over the punctured disk. An interpolation argument gives uniform Hörmander-Tian type estimates along the flows. We also establish uniform diameter bounds and Gromov–Hausdorff precompactness for this family of flows. In particular, we obtain an alternative proof of the diameter estimate by Li-Tosatti.

Type I estimates for the Kahler-Ricci flow I

Wed, 30 Sep 2026 00:00:00 GMT

Wangjian Jian, Jian Song — We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.

Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

Thu, 17 Sep 2026 00:00:00 GMT

Wangjian Jian, Jian Song — The analytic minimal model program proposed in seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.

Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

Wed, 16 Sep 2026 00:00:00 GMT

Tongxin Xu, Zhenlei Zhang — We prove that any finite-time collapsing Kähler-Ricci flow on compact Kähler surfaces develops a Type I singularity. Together with the previous results, this implies that any finite time singularity of the Kähler-Ricci flow on compact Kähler surfaces is of Type I.

Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

Tue, 15 Sep 2026 00:00:00 GMT

Charles Cifarelli, Ronan Conlon, Max Hallgren, Junsheng Zhang — For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.

Finite-Time Singularities of the Kähler–Ricci Flow on a ℂP^m-Bundle over a Product of Kähler–Einstein Manifolds

Mon, 07 Sep 2026 00:00:00 GMT

Yifan Xiao — In this paper, we study the Kähler–Ricci flow on ℂP^m-bundles over a product of Kähler–Einstein manifolds, starting from an initial metric with Calabi symmetry. We prove that every finite-time singularity arising along the flow must be of Type I.

On Miyaoka-Yau Inequalities and Weil-Petersson Metrics

Sun, 06 Sep 2026 00:00:00 GMT

Alexander Bednarek — We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is n-1, we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.

Singularity Models of Finite-Time Kähler-Ricci Flows

Thu, 03 Sep 2026 00:00:00 GMT

Frederick Tsz-Ho Fong, Hung Tran — We study the singularity type and models of the Kähler–Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler–Einstein manifolds N := N₁ × ⋯ × Nᵣ, with metric constructed using the ansatz considered in, et. al. In the earlier work by the authors, we considered the "two-bolt" case where both ends of the foliation close with the "bolt" N. In this article, we continue our work on the more subtle "nut-bolt" and "two-nut" cases. The former has one end of the interval closes with a nut-type collapse (i.e. N' := N₂ × ⋯ × Nᵣ) and the other with a bolt (i.e. N). The compactification M is then a ℂP}^(m+1)-bundle over N'. The "two-nut" case is one that both ends close with nut-type collapses, necessarily two of the Nᵢ's must be ℂP}^(m₀) and ℂP}^(m_ℓ), and the compactification M is a ℂP}^(m₀+m_ℓ+1)-bundle over ∏_(k≥ 3)Nₖ. We proved that in all "two-bolt", "nut-bolt" and "two-nut" caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be (Σ^(m+1), g_Σ(t)) × (ℂ^(k), flat) with m, k ≥ 0, where Σ is one of the following: ℂP}^(m+1), Tot(𝓛^(⊕(m+1))), or a projectivization ℙ(𝓞^(⊕(m₀+1)) ⊕ 𝓛^(⊕(m_ℓ+1))) with m₀ + m_ℓ = m, and 𝓛 is a line bundle over the product of some of the N₁, ⋯, Nᵣ factors. The metric g_Σ(t) is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.

Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles

Wed, 02 Sep 2026 00:00:00 GMT

Wangjian Jian, Jian Song — We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle Xⁿ→ Y^m, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as (C^m,g_(rm E),J₀)×(Z',d',J'). If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to √(T-t). Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder C^m×PP¹.

Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

Tue, 01 Sep 2026 00:00:00 GMT

Tongxin Xu, Zhenlei Zhang — We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker ℙ¹× ℂ. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.

Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow

Mon, 31 Aug 2026 00:00:00 GMT

Xiangsen Qin — Di Nezza–Guedj–Lu asked whether local Arnold multiplicities under maximal weak twisted Kähler–Ricci flow with fixed cohomology class satisfy λ(φₜ,x)=max\λ(φ₀,x)−t,0\. We construct counterexamples with analytic initial singularities in every dimension n≥2. On the Hirzebruch surface Fₑ, e≥2, the initial current [S]+(e+1)[F] yields λ(φₜ,x)=1/2-t/e for x∈ S∖ F and 0

Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

Wed, 19 Aug 2026 00:00:00 GMT

Longteng Chen, Max Hallgren, Lucas Lavoyer — 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.

Finite Time Type I Singularities of the Kähler Ricci Flow

Tue, 18 Aug 2026 00:00:00 GMT

Tongxin Xu, Zhenlei Zhang — We prove the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow. More precisely, for any compact Kähler manifold Y and its blow-up π:BlₚY⟶ Y, if [ω₀]−Tc₁(M)=π^*[ω_Y], then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker Tot(𝓞_(ℙⁿ⁻¹)(−1)).

Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

Tue, 28 Jul 2026 00:00:00 GMT

Yu Li, Junsheng Zhang — We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an S¹-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.

Geometric smoothing by the Kähler-Ricci Flow

Thu, 09 Jul 2026 00:00:00 GMT

Eleonora Di Nezza, Vincent Guedj, Chinh Hoang Lu — We study the geometric regularization of a positive closed current by the (twisted) Kähler-Ricci flow on a compact Kähler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete Kähler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.

Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold

Tue, 16 Jun 2026 00:00:00 GMT

Lei Zhang, Zhenlei Zhang — This paper is concerned with a class of the long time Kähler-Ricci flow on a compact Kähler manifold. It is shown that the uniform μ-entropy or uniform Sobolev inequality along the normalized Kähler-Ricci flow with semiample canonical bundle. As a consequence, we prove that the scalar curvature of the Kähler metrics along the normalized Kähler-Ricci flow converge to negative Kodaira dimension of the compact Kähler manifold.

Kähler-Ricci Flow: from divisors to cusps

Sun, 14 Jun 2026 00:00:00 GMT

Eleonora Di Nezza, Vincent Guedj, Chinh H. Lu — We study the geometric regularization of positive closed currents by the Kähler-Ricci flow on compact Kähler manifolds. In a previous work of ours, it was shown that the Kähler-Ricci flow immediately smoothes out such a current when it has zero Lelong numbers. We study here the case when T₀ has divisorial singularities, showing that the flow gradually replaces the latter by Poincaré type ones, providing an approximation of T₀ by complete Kähler metrics with bounded curvature in a Zariski open set.

Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow

Sun, 07 Jun 2026 00:00:00 GMT

Longteng Chen — In this paper, we investigate the relationship between the long time behaviour of solutions to the Kahler-Ricci flow on an asymptotically conical gradient Kahler-Ricci expander and the asymptotic behaviour of their initial data at spatial infinity.

Scalar Curvature, Volumes and the Bergman Kernel

Sun, 31 May 2026 00:00:00 GMT

Bo-Yong Chen, Yuanpu Xiong, Liyou Zhang — Motivated by the works of Gromov and LeBrun in Riemannian geometry, we study the analogous phenomena in complex geometry. We first show that both ∫_M |S_C⁻(g)|ⁿdV_g and rm vol_g(M) (normalized by S_C(g)≥ −1) are bounded below by ((nπ)ⁿ)/(n!)CanVol(M) for any Hermitian metric g on a compact complex n-manifold M. Here S_C denotes the Chern scalar curvature, S_C⁻=max\−S_C,0\ and rm CanVol(M) is the canonical volume of M, i.e., the volume of the canonical line bundle K_M. Moreover, if rm vol_g(M)=((nπ)ⁿ)/(n!)CanVol(M) holds for some Kähler metric with S_C≥ −1, then it has to be the Kähler-Einstein metric of negative scalar curvature. The completely new phenomenon is that if M is a compact Kähler manifold such that K_M is nef, then rm MinVol_C(M)=𝓘_C(M)=𝓘_C⁻(M)=((nπ)ⁿ)/(n!)CanVol(M), where rm MinVol_C(M) is the infimum of rm vol_g(M) with S_C(g)≥ −1 and 𝓘_C⁻(M)=inf_g ∫_M |S_C⁻(g)|ⁿdV_g, 𝓘_C(M)=inf_g ∫_M |S_C(g)|ⁿdV_g. It remains unknown whether the nef condition is superfluous. The answer is positive when M is obtained by blowing up a finite number of points from a projective manifold with big and nef canonical line bundle. The arguments are based on the asymptotic behaviour of the Bergman kernel of mK_M as m→ ∞, the theory of Kähler-Ricci flow and singular Kähler-Einstein metric, as well as a very delicate gluing technique, using the Burns-Simanca metric.

Singular Kähler–Ricci shrinkers and polarized Fano fibrations

Sun, 24 May 2026 00:00:00 GMT

Max Hallgren, Junsheng Zhang — We prove that any singular Kähler–Ricci shrinker X arising as a noncollapsed limit of Kähler–Ricci flows admits a natural structure of a polarized Fano fibration. We also show that it is simply connected, has unique tangent cones at every point, and is a smooth orbifold outside a subset of complex codimension at least three. As an application, we prove a new long-time pseudolocality theorem for almost-selfsimilar Kähler–Ricci flows.

Complex normalizing flows can almost be information Kähler-Ricci flows

Mon, 20 Apr 2026 00:00:00 GMT

Andrew Gracyk — We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly Kähler-Ricci. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the ensemble of Wirtinger Jacobians. The Ricci curvature of a Kähler manifold is the second order mixed Wirtinger partial derivative of the log of the local density of the volume form. Therefore, we reconcile these two facts by drawing forth the connection that the log determinant used in the complex normalizing flow matches a Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial information metric under an augmented Jacobian and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, or more closely a Kähler cross-entropy Hessian. This recovers a Kähler-Ricci flow variation up to a time derivative and expectation, or an average-valued Kähler-Einstein flow. Using this framework, we establish other relevant results, attempting to bridge the statistical and ordinary behaviors of the complex normalizing flow to the geometric features of our derived Kähler flow.

Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature

Sun, 19 Apr 2026 00:00:00 GMT

Jintao Ye, Xiaohua Zhu — We establish a Petersen-Wei type relative volume comparison theorem for weighted Riemannian manifolds under both L^p-bounds of the Bakry-Émery Ricci curvature and the gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity of Kähler-Ricci flow established in a recent work of Tian-Zhang-Zhang-Zhu-Zhu with improved estimate for error term.

Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities

Sun, 05 Apr 2026 00:00:00 GMT

Longteng Chen, Max Hallgren, Lucas Lavoyer — Let (Y,g₀) be a compact analytic space with a finite number of singular points, where the metric at each singular point is modelled on a Kähler cone with smooth canonical model. We show that the Kähler-Ricci flow with such initial data satisfies a C/t curvature bound, and that the flow near each singular point is modelled on the unique Kähler-Ricci expander asymptotic to the corresponding cone. Our motivation is to give a geometric description of the Kähler–Ricci flow emerging from singularities arising in the analytic minimal model program.

Higher-order Ricci estimates along immortal Kähler-Ricci flows

Wed, 25 Mar 2026 00:00:00 GMT

Wenrui Kong — We study higher-order curvature estimates along Kähler-Ricci flows on compact Kähler manifolds of intermediate Kodaira dimension. We prove that away from singular fibers, the Ricci curvature is uniformly bounded in C¹, the Laplacian of the Ricci curvature in C⁰, and the scalar curvature in C². We identify a geometric obstruction to higher-order curvature bounds, whose non-vanishing causes a specific third-order derivative of the Ricci curvature to blow up at rate e^(t/2). Uniform C^k bounds for every k hold for the Ricci curvature in the isotrivial case, and for the full Riemann curvature in the torus-fibered case.