https://neckpinch.com/topics/minimal-model-program/New arXiv papers on analytic minimal model program, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.22941arxiv:2610.22941Fri, 16 Oct 2026 00:00:00 GMTWangjian 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.Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometryScalar curvatureLimit spaces & RCDAnalytic minimal model program https://arxiv.org/abs/2610.21934arxiv:2610.21934Thu, 15 Oct 2026 00:00:00 GMTFrederick 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.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryScalar curvatureLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2610.05643arxiv:2610.05643Mon, 05 Oct 2026 00:00:00 GMTYu Li, Junsheng Zhang — We prove a uniform scalar curvature bound, depending only on the dimension, for Kähler–Ricci shrinkers. We also prove that every noncompact Kähler–Ricci shrinker has a unique tangent cone at infinity, naturally homeomorphic to the base of its polarized Fano fibration. For shrinkers with maximal volume growth, this cone is isometric to the metric completion of the regular part equipped with the intrinsic distance induced by the limiting Kähler metric.Ricci solitonsKähler geometryScalar curvatureBoundaries & non-compactAnalytic minimal model program https://arxiv.org/abs/2610.03592arxiv:2610.03592Fri, 02 Oct 2026 00:00:00 GMTCarlos Esparza, Junsheng Zhang — For noncompact smooth Kähler-Ricci shrinkers, we prove a Matsushima theorem identifying the centralizer of the soliton field with the complexification of its Killing centralizer, uniqueness up to biholomorphism on a fixed complex manifold, and K-polystability of the associated polarized Fano fibration.Ricci solitonsKähler geometryStability & uniquenessBoundaries & non-compactKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2609.39620arxiv:2609.39620Wed, 30 Sep 2026 00:00:00 GMTWangjian 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.Ricci flowKähler–Ricci flowSingularitiesKähler geometryScalar curvatureAnalytic minimal model program https://arxiv.org/abs/2609.24884arxiv:2609.24884Mon, 21 Sep 2026 00:00:00 GMTTristan C. Collins, Genggeng Huang, Freid Tong, Yulun Xu — We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.Ricci solitonsKähler–Ricci solitonsKähler geometryAnalytic minimal model program https://arxiv.org/abs/2609.20513arxiv:2609.20513Thu, 17 Sep 2026 00:00:00 GMTWangjian 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.Ricci flowKähler–Ricci flowEinstein metricsSingularities(Non)collapsingKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2609.06451arxiv:2609.06451Sun, 06 Sep 2026 00:00:00 GMTAlexander 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.Ricci flowKähler–Ricci flowKähler geometryStability & uniquenessKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2609.02878arxiv:2609.02878Wed, 02 Sep 2026 00:00:00 GMTWangjian 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.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometryAnalytic minimal model program https://arxiv.org/abs/2608.19152arxiv:2608.19152Wed, 19 Aug 2026 00:00:00 GMTLongteng 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.Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometryWeak & singular flowsBoundaries & non-compactAnalytic minimal model program https://arxiv.org/abs/2608.17458arxiv:2608.17458Tue, 18 Aug 2026 00:00:00 GMTTongxin 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)).Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometryAnalytic minimal model program https://arxiv.org/abs/2608.10953arxiv:2608.10953Tue, 11 Aug 2026 00:00:00 GMTTongxin Xu, Zhenlei Zhang — In this paper, we study compactness, rigidity, and related geometric properties of complete Kähler Ricci shrinkers through the polarized Fano fibration structure.Ricci solitonsKähler geometryLimit spaces & RCDStability & uniquenessAnalytic minimal model program https://arxiv.org/abs/2607.25181arxiv:2607.25181Tue, 28 Jul 2026 00:00:00 GMTTomoyuki Hisamoto, Masataka Iwai — We establish the following Miyaoka-Yau inequality for any n-dimensional klt Fano variety X, possibly K-unstable, in terms of its delta invariant: (2(n+1)c₂(X)−n c₁(X)²)· c₁(X)ⁿ⁻² ≥ −n (1-min\1,δ(X)\)² · c₁(X)ⁿ. Furthermore, inspired by recent work of Inoue and Hallam-Lahdili, we formulate and prove an equivariant version of this inequality in the soliton setting, in which the Chern classes and the delta invariant are replaced by the equivariant Chern classes and the weighted delta invariant, respectively. As a consequence, we obtain the equivariant Miyaoka-Yau inequality for every smooth Fano manifold admitting a Kähler-Ricci soliton.Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniquenessKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2606.17402arxiv:2606.17402Tue, 16 Jun 2026 00:00:00 GMTLei 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.Ricci flowKähler–Ricci flowKähler geometryScalar curvatureAnalytic minimal model program https://arxiv.org/abs/2606.01153arxiv:2606.01153Sun, 31 May 2026 00:00:00 GMTBo-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.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2605.25213arxiv:2605.25213Sun, 24 May 2026 00:00:00 GMTMax 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.Ricci flowKähler–Ricci flowRicci solitons(Non)collapsingKähler geometryAnalytic minimal model program https://arxiv.org/abs/2605.22347arxiv:2605.22347Thu, 21 May 2026 00:00:00 GMTHaoyuan Sun — We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.Ricci flowGeneralized & G₂ flows(Non)collapsingKähler geometryLimit spaces & RCDMinimal & CMC surfacesAnalytic minimal model program https://arxiv.org/abs/2605.20852arxiv:2605.20852Wed, 20 May 2026 00:00:00 GMTShu-Cheng Chang, Yingbo Han, Chien Lin, Chin-Tung Wu — In this paper, we first confirm the Hamilton-Tian conjecture for the Sasaki-Ricci flow in a compact transverse Fano quasi-regular Sasakian 5-manifold with klt foliation singularities. Secondly, we derive the compactness theorem of Sasaki-Ricci solitons on transverse Fano quasi-regular Sasakian 5-manifolds. Then,by the second Sasakian structure theorem, we confirm the Hamilton-Tian conjecture for a compact transverse Fano Sasakian 5-manifold. With its applications, we show that the gradient Sasaki-Ricci soliton orbifold metric on a compact Sasakian 5-manifold is Sasaki-Einstein if M is transverse K-stable.Ricci flowRicci solitonsSingularitiesKähler geometryLimit spaces & RCDAnalytic minimal model programSasakian & contact geometry https://arxiv.org/abs/2604.04710arxiv:2604.04710Mon, 06 Apr 2026 00:00:00 GMTHaoyuan Sun — We establish uniform diameter estimates and volume non-collapsing estimates for the Chern-Ricci flow on smooth Hermitian minimal models of general type, assuming the initial metric is Kähler in a neighborhood of the null locus of the canonical bundle. This yields subsequential Gromov-Hausdorff convergence, partially resolving a conjecture of Tosatti and Weinkove. When the underlying manifold is Kähler, we further prove the uniqueness of the limit space. Analytically, we overcome the difficulties posed by non-Kähler torsion in the Green's formula by exploiting our local Kähler assumption, successfully adapting recent estimates of Kähler Green's function to the Hermitian setting. To prove the uniqueness of the limit, we introduce Perelman's reduced length to the Chern-Ricci flow. By establishing a uniform Chern scalar curvature bound and an almost monotonicity formula for the reduced volume, we deduce an almost-avoidance principle for the singular set, allowing us to effectively compare the flow distance with the canonical limit distance.Ricci flowGeneralized & G₂ flowsReduced volume(Non)collapsingKähler geometryScalar curvatureLimit spaces & RCDStability & uniquenessAnalytic minimal model program https://arxiv.org/abs/2604.04223arxiv:2604.04223Sun, 05 Apr 2026 00:00:00 GMTLongteng 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.Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometryAnalytic minimal model program https://arxiv.org/abs/2603.24380arxiv:2603.24380Wed, 25 Mar 2026 00:00:00 GMTWenrui 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.Ricci flowKähler–Ricci flowSingularitiesKähler geometryScalar curvatureLong-time existence & convergenceAnalytic minimal model program https://arxiv.org/abs/2602.19913arxiv:2602.19913Mon, 23 Feb 2026 00:00:00 GMTMan-Chun Lee, Valentino Tosatti, Junsheng Zhang — We show that the normalized Kähler-Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model, as conjectured by Song-Tian's analytic mimimal model program.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryLimit spaces & RCDLong-time existence & convergenceKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2512.21910arxiv:2512.21910Fri, 26 Dec 2025 00:00:00 GMTAlexander Bednarek — This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of C⁻¹(T-t)^(n-m) Ω≤ ω(t)ⁿ≤ C(T-t)^(n-m)Ω. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate √(T-t). Furthermore, several precise C⁰-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion.Ricci flowKähler–Ricci flowEinstein metricsSingularitiesKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2512.21904arxiv:2512.21904Fri, 26 Dec 2025 00:00:00 GMTAlexander Bednarek — This is the first of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem from a compact Kähler manifold X endowed with an ample, rational line bundle L and non-nef canonical line bundle K_X, we construct a (1,1)-form on the regular part of the base analytic variety which is related to the Weil-Petersson metric. It is also proven that the singular Kähler metric constructed by Zhang, Zhang, on the base analytic variety satisfies a twisted Kähler-Einstein equation involving this (1,1)-form and, for a submersion, that the Chern classes of X and the base manifold decompose in terms of this (1,1)-form.Ricci flowKähler–Ricci flowEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2512.03323arxiv:2512.03323Wed, 03 Dec 2025 00:00:00 GMTCharles Cifarelli, Carlos Esparza — Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration (π: X → Y, ξ). In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric g implies K-polystability of (π: X → Y, ξ), in the case that the Ricci curvature of g decays at infinity. As an application, we give a non-existence result: if M is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space X of the cube root of K_M is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.Ricci solitonsSingularitiesSurfacesKähler geometryStability & uniquenessBoundaries & non-compactKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2506.14671arxiv:2506.14671Tue, 17 Jun 2025 00:00:00 GMTYuji Odaka — We revisit the recent theory of Sun-Zhang on general Fano fibration (germs) which emerged from the study of non-compact Kahler-Ricci soliton metrics, primarily from an algebro-geometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction ("bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibration.Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compactAnalytic minimal model program https://arxiv.org/abs/2503.05838arxiv:2503.05838Thu, 06 Mar 2025 00:00:00 GMTCarlos Esparza — We show that smooth polarized Fano fibrations have no nontrivial finite covers. Using results by Sun-Zhang and Wylie, it follows that shrinking Kähler-Ricci solitons are simply-connected.Ricci solitonsKähler–Ricci solitonsKähler geometryAnalytic minimal model program https://arxiv.org/abs/2502.13521arxiv:2502.13521Wed, 19 Feb 2025 00:00:00 GMTCarlos Esparza — We show that, up to biholomorphism, a given noncompact complex manifold only admits one shrinking gradient Kähler-Ricci soliton with Ricci curvature tending to zero at infinity. Our result does not require fixing the asymptotic data of the metric, nor fixing the soliton vector field. The method used to prove the uniqueness of the soliton vector field can be applied more widely, for example to show that conical Calabi-Yau metrics on a given complex manifold are unique up to biholomorphism. We also use it to prove that if two polarized Fano fibrations, as introduced by Sun-Zhang, are biholomorphic, then they are isomorphic as algebraic varieties.Ricci solitonsKähler–Ricci solitonsRicci-flat & Calabi–YauKähler geometryStability & uniquenessBoundaries & non-compactAnalytic minimal model program https://arxiv.org/abs/2502.09825arxiv:2502.09825Thu, 13 Feb 2025 00:00:00 GMTYifan Chen, Shih-Kai Chiu, Max Hallgren, Gábor Székelyhidi, Tat Dat Tô, Freid Tong — We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs, and an analogous result holds for Kähler-Ricci solitons. In addition we show that if a singular Kähler-Einstein metric can be approximated by smooth metrics on a resolution whose Ricci curvature has negative part that is bounded uniformly in L^p for p > (2n-1)/n, then the metric defines an RCD space.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2501.12759arxiv:2501.12759Wed, 22 Jan 2025 00:00:00 GMTJohn Lott — We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.Ricci flow4-manifoldsLong-time existence & convergenceMinimal & CMC surfacesAnalytic minimal model program