Kähler–Ricci flow

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 $C^2$-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.

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 $C^0$ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

Let $X$ be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class $[\omega ]$, Jian–Shi–Song proved that there exists a unique cscK metric in the class $2\pi c_1(K_X)+\epsilon [\omega ]$ when $\epsilon \in (0,{\epsilon }_0]$, using the properness criterion of Chen–Cheng. We prove that the cscK metrics in $2\pi c_1(K_X)+\epsilon [\omega ]$ converge, as $\epsilon \downarrow 0$, 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 $X$.

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 $r\lesssim T-t$ 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 $W_1$-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances $d_t$ (induced by $\omega (t)$) 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.

We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold $(M,J,\mathbb{T})$, and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial $\mathbb{T}$-invariant generalized Kähler metric with its symplectic form in $2\pi c_1(M,J)$, we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.

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.

We prove a uniform $L^{\infty }$-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.

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.

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.

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.

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.

In this paper, we study the Kähler–Ricci flow on $\mathbb{C}{P}^m$-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.

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.

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 $N:=N_1\times \cdots \times N_r$, 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. $N^{\prime }:=N_2\times \cdots \times N_r$) and the other with a bolt (i.e. $N$). The compactification $\hat{M}$ is then a ${\mathbb{CP}}^{m+1}$-bundle over $N^{\prime }$. The "two-nut" case is one that both ends close with nut-type collapses, necessarily two of the $N_i$'s must be ${\mathbb{CP}}^{m_0}$ and ${\mathbb{CP}}^{m_{\ell }}$, and the compactification $\hat{M}$ is a ${\mathbb{CP}}^{m_0+m_{\ell }+1}$-bundle over ${\prod }_{k\ge 3}{N}_k$. 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 $({\Sigma }^{m+1},g_{\Sigma }(t))\times ({\mathbb{C}}^k,\text{flat})$ with $m,k\ge 0$, where $\Sigma$ is one of the following: ${\mathbb{CP}}^{m+1}$, $\text{Tot}({\mathcal{L}}^{\oplus (m+1)})$, or a projectivization $\mathbb{P}({\mathcal{O}}^{\oplus (m_0+1)}\oplus {\mathcal{L}}^{\oplus (m_{\ell }+1)})$ with $m_0+m_{\ell }=m$, and $\mathcal{L}$ is a line bundle over the product of some of the $N_1,\cdots ,N_r$ factors. The metric $g_{\Sigma }(t)$ is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.

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 $X^n\to Y^m$, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as $({\text{C}}^m,g_{\mathrm{E}},J_0)\times (Z^{\prime },d^{\prime },J^{\prime })$. 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 $\sqrt{T-t}$. Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder ${\text{C}}^m\times {\text{PP}}^1$.

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 ${\mathbb{P}}^1\times \mathbb{C}$. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.

Di Nezza–Guedj–Lu asked whether local Arnold multiplicities under maximal weak twisted Kähler–Ricci flow with fixed cohomology class satisfy $\lambda ({\phi }_t,x)=\max \{\lambda ({\phi }_0,x)-t,0\}$. We construct counterexamples with analytic initial singularities in every dimension $n\ge 2$. On the Hirzebruch surface ${\mathbb{F}}_e$, $e\ge 2$, the initial current $[S]+(e+1)[F]$ yields $\lambda ({\phi }_t,x)=1/2-t/e$ for $x\in S\setminus F$ and $0<t<e/2$, where $S$ is the negative section and $F$ a fibre. Products with smooth stationary factors give higher-dimensional examples. The Zariski negative part of $\{S\}+\{F\}$, the reduced support class, forces extra divisor mass. For pure divisorial data with simple normal crossings support, a smooth semipositive representative of the reduced support class's positive part gives exact short-time formulas. On one surface we construct flows with identical initial current germs, background data and global initial Arnold multiplicity, but different local rates.

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.

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 $\pi :{\mathrm{Bl}}_{p}Y⟶Y$, if $[{\omega }_0]-T{c}_1(M)={\pi }^{\ast }[{\omega }_Y]$, then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker $\mathrm{Tot}({\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1))$.

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 $S^1$-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.