Kähler geometry

The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.

We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.

We prove that every extremal Kähler–Ricci soliton on a Fano manifold is Kähler–Einstein. This solves the problem of Calamai and Petrecca in full generality.

We prove linear stability of all steady and expanding gradient Kähler-Ricci solitons. In the expanding case, we prove strict linear stability under very general assumptions. In particular, every asymptotically conical expanding gradient Kähler-Ricci soliton is strictly linearly stable.

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.

In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete gradient shrinking Kähler-Ricci solitons in complex dimension two, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.

Let $\lambda (u,x)$ be the local Arnold multiplicity of a quasi-plurisubharmonic function $u$. Di Nezza–Guedj–Lu asked whether every maximal weak solution ${\phi }_t$ of the twisted Kähler–Ricci flow satisfies $\lambda ({\phi }_t,x)=\max \{\lambda ({\phi }_0,x)-t,0\}$. We give counterexamples on the Hirzebruch surface ${\mathbb{F}}_e={\mathbb{P}}_{{\mathbb{P}}^1}({\mathcal{O}}_{{\mathbb{P}}^1}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-e))$, $e\ge 2$. Let $S$ be its negative section and $F_1,\dots ,F_k$ be distinct fibres. If $a,b_i>0$, ${\sum }_i{b}_i>ea$, and the initial current is $a[S]+{\sum }_i{b}_i[F_i]$, then $\lambda ({\phi }_t,x)=a-\min \{k/e,1\}t$ for $x\in S\setminus {\bigcup }_i{F}_i$ and $0<t<\min \{a,b_1,\dots ,b_k\}$. Thus the formula fails for $k<e$; the slower decay is forced by the Zariski negative part of the residual class. Under explicit SNC and positivity hypotheses, we also prove an exact surface formula for pure divisorial data. Consequences include nonlocality with fixed background data and explicit multiplier ideals. Products with smooth factors give counterexamples in every complex dimension $n\ge 2$.

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 $\mathrm{Tot}({\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1))$.

Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-Einstein manifold in dimension $5$ and $7$. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with $S^1$.

We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively $C^{2,\alpha }$-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder $h^{2,\alpha }$ neighborhood, a fixed positive-time restart yields the marked first-profile coordinate ${\mathfrak{A}}_1={\lambda }_{\infty }^{-{\gamma }_1}{V}_{\infty }\in E_1$. This amplitude is a split $C^1$ submersion and locally the projection onto $E_1$. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.

We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton $(M,g,J,f)$ and a real-valued pluriharmonic function $u$, we investigate conditions under which $u$ must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that $u$ is constant whenever $${\int }_M|\nabla u{|}^p\mathrm{d}v<\infty$$ for some $0<p<\infty$. In the shrinking case, we prove the same conclusion for $0<p\le 2$. Finally, we construct a complete Kähler example showing that the extension to the range $0<p<1$ relies essentially on the soliton structure and does not hold on general complete Kähler manifolds.

We show that any complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient Kähler-Ricci soliton on such a resolution. This confirms a special case of the uniqueness part of a conjecture of Song-Zhang. Some other consequences are also discussed.

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.

We establish the following Miyaoka-Yau inequality for any $n$-dimensional klt Fano variety $X$, possibly K-unstable, in terms of its delta invariant: $$\left(2(n+1){\hat{c}}_2(X)-n{c}_1(X{)}^2\right)\cdot c_1(X{)}^{n-2}\ge -n{\left(1-\min \{1,\delta (X)\}\right)}^2\cdot c_1(X{)}^n.$$ 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.

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.

In this paper, we first propose two different quantities on graphs, namely local entropy and cohesion, then design two corresponding flows for edge weights: the local entropy flow and the cohesion flow. We establish the global existence and uniqueness of solutions for both flows and investigate their asymptotic behaviors, including the case that the limit goes to positive infinity. Moreover, they can be applied to fundamental network analysis tasks, including community detection and node classification. Empirical evaluations demonstrate that our method achieves performance competitive with Ollivier Ricci flow and Lin-Lu-Yau Ricci flow on benchmark network analysis tasks. In experimental scenarios, we first apply the cohesion flow to evolve the edge weights of the graph, and then apply the local entropy flow to further update the resulting weighted graph. Both flows are computationally efficient, leading to a significant reduction in overall computational cost and improved scalability.

We study Riemann solitons and $\eta$-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds. Under the Einstein field equations with cosmological constant and perfect fluid assumptions, explicit formulas for the soliton parameter are derived, yielding criteria for shrinking, steady, and expanding behaviors. Several physically relevant models, including dark fluid, stiff matter, dust, and radiation, are analyzed. We show that Bochner-flat Lorentzian Kähler spacetimes are Einstein and investigate the resulting geometric and dynamical consequences. In the context of generalized Robertson–Walker spacetimes, we obtain constraints on the warping function and classify soliton solutions. Global properties such as geodesic completeness, singularity formation, and stability are also examined.

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 $\mu$-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.

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 $T_0$ has divisorial singularities, showing that the flow gradually replaces the latter by Poincaré type ones, providing an approximation of $T_0$ by complete Kähler metrics with bounded curvature in a Zariski open set.

The analysis of singularities in scalar curvature is a classical problem. In this survey, we investigate the behavior of scalar curvature under several geometric flows, with a focus on three specific cases: the Ricci flow, the Kähler-Ricci flow coupled with $(1,1)$-forms, and the Laplacian flow.

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.

Motivated by the works of Gromov and LeBrun in Riemannian geometry, we study the analogous phenomena in complex geometry. We first show that both ${\int }_M|S_C^{-}(g){|}^{n}d{V}_g$ and ${\mathrm{vol}}_g(M)$ (normalized by $S_C(g)\ge -1$) are bounded below by $\frac{(n\pi {)}^n}{n!}\mathrm{CanVol}(M)$ 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 $\mathrm{CanVol}(M)$ is the canonical volume of $M$, i.e., the volume of the canonical line bundle $K_M$. Moreover, if ${\mathrm{vol}}_g(M)=\frac{(n\pi {)}^n}{n!}\mathrm{CanVol}(M)$ holds for some Kähler metric with $S_C\ge -1$, 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 ${\mathrm{MinVol}}_C(M)={\mathcal{I}}_C(M)={\mathcal{I}}_C^{-}(M)=\frac{(n\pi {)}^n}{n!}\mathrm{CanVol}(M)$, where ${\mathrm{MinVol}}_C(M)$ is the infimum of ${\mathrm{vol}}_g(M)$ with $S_C(g)\ge -1$ and ${\mathcal{I}}_C^{-}(M)={\inf }_g{\int }_M|S_C^{-}(g){|}^{n}d{V}_g$, ${\mathcal{I}}_C(M)={\inf }_g{\int }_M|S_C(g){|}^{n}d{V}_g$. 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 $m{K}_M$ as $m\to \infty$, the theory of Kähler-Ricci flow and singular Kähler-Einstein metric, as well as a very delicate gluing technique, using the Burns-Simanca metric.

The Fefferman–Szegő metric $g_{\mathrm{FS}}^{\Omega }$ on a $C^{\infty }$-smooth bounded strongly pseudoconvex domain $\Omega \subset {\mathbb{C}}^n$ is an invariant metric defined via the Fefferman surface measure. For this metric, we first establish the vanishing of its $L^2$-Dolbeault cohomology outside the middle degree: $\dim H_2^{p,q}(\Omega )=0$ if $p+q\ne n$, while $\dim H_2^{p,q}(\Omega )=\infty$ if $p+q=n$. We also prove that the metric has $C^{\infty }$-bounded geometry. Using this analytic property, we obtain several rigidity results. In particular, if the Fefferman–Szegő metric is a gradient Kahler–Ricci soliton, then $\Omega$ is biholomorphic to the unit ball ${\mathbb{B}}^n$. Moreover, if the metric has constant scalar curvature, then it is Einstein, and again $\Omega$ is biholomorphic to ${\mathbb{B}}^n$. We also give a Ramadanov-type criterion in terms of the Fefferman–Szegő invariant function. Finally, in dimension $n=2$, assuming the existence of a Kahler immersion into a finite-dimensional ball that maps boundary to boundary transversally, we show that the logarithmic term of the Fefferman–Szegő kernel vanishes to infinite order. Consequently, the boundary is locally spherical; if, in addition, $\Omega$ is simply connected, then $\Omega$ is biholomorphic to ${\mathbb{B}}^2$.

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.

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.

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.

The notion of weighted extremal Kähler metrics extends the classical notion of Calabi's extremal Kähler metrics, but includes many well-studied objects in Kähler geometry such as Kähler-Ricci solitons and Sasaki-Einstein metrics. In this paper, after explaining how this notion grew out, we will try to survey recent works concerning the YTD conjecture on weighted extremal Kähler metrics.

This paper studies a non-trivial gradient Kähler-Ricci soliton, of complex dimension $n$, with an isometry group of dimension at least $n^2-1$. We show that the isometry group acts by cohomogeneity one and, consequently, admits a special ansatz involving a Sasakian model. In complex dimension two, we can actually say more: namely, that every such soliton has maximal symmetry; that is, the isometry group is exactly of dimension $2^2$. In addition, we prove that, if the isometry group acts by cohomogeneity one on a non-trivial gradient Ricci soliton (not necessarily Kähler), the potential function is invariant by the action.

We establish a general result ensuring a $C^1$ a priori bound for smooth curves of Hermitian metrics. As a main application, we obtain a new regularity result for Hermitian curvature flows, and in particular for the second Chern-Ricci flow.

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.

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.

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.

Let $(Y,g_0)$ 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.

We study the stability and Hölder continuity of solutions to degenerate complex Monge–Ampère equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when the reference form is the pullback of a Hermitian metric. As a consequence, we establish a uniform diameter bound for the twisted Chern–Ricci flow.

We construct an example of an asymptotically conical (AC) non-Kähler expanding gradient Ricci soliton that has a Kähler tangent cone at infinity. This yields an example of a Kähler cone that can be desingularised by a smooth AC expanding gradient Ricci soliton but not by a smooth AC expanding gradient Kähler–Ricci soliton.

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 $C^1$, the Laplacian of the Ricci curvature in $C^0$, and the scalar curvature in $C^2$. 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.

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.

In this paper, we study the Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds whose initial metric is constructed by the ansatz used in works by M. Wang et. al. We prove that the ansatz is preserved along the Ricci flow. Furthermore, in the Kähler case, we proved that Type I finite-time singularity must occur under such an ansatz.

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 $C^{-1}(T-t{)}^{n-m}\Omega \le \omega (t{)}^n\le C(T-t{)}^{n-m}\Omega$. 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 $\sqrt{T-t}$. Furthermore, several precise $C^0$-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.

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.

We show that any toric asymptotically conical shrinking gradient Kähler-Ricci soliton on an anti-canonically polarised resolution of a Kähler cone satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity.

We establish global existence of the pluriclosed flow with arbitrary initial data on Oeljeklaus-Toma manifolds, and Gromov-Hausdorff convergence of blowdown limits to a torus under natural conjectural bounds on the flow at infinity. In the case of generalized Kähler-Ricci flow we prove refined a priori estimates in support of these conjectural bounds.

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.

We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of and, via recent work the on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted $L^2$-spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension $4$. We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.

We prove a weaker version of the transcendental base-point freeness on compact Kähler manifolds. As a consequence, we derive the diameter lower bound for finite time singularities of Kähler-Ricci flow with non-Fano initial data.

Given a compact Kähler manifold $X$ and a closed, positive $(1,1)$-current $T$ on $X$, we find sufficient conditions for $T$ to induce a metric structure $(X,d_T)$ which is the Gromov-Hausdorff limit of compact Kähler manifolds either in a "static" way or at time zero of smooth Kähler-Ricci flows. In dimension $1$ we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a Kähler-Ricci flow.

We consider noncompact complete Kähler manifolds with nonnegative bisectional curvature. Our main results are: 1. Precise relations among refined minimal degree of polynomial growth holomorphic functions and holomorphic volume forms, $\mathrm{AVR}$ (asymptotic volume ratio) and $\mathrm{ASCD}$ (average of scalar curvature decay) are established. 2. The Lyapunov asymptotic behavior of the Kähler-Ricci flow can be described in terms of polynomial growth holomorphic functions. This provides a unifying perspective that bridges the two distinct proofs of Yau's uniformization conjecture by Liu and Chau-Lee-Tam. These resolve two conjectures made by Yang.

We prove that on ALF $n$-manifolds with $n\ge 4$ the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's $\lambda$-functional, we define a renormalized functional ${\lambda }_{\mathrm{ALF}}$ whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF $4$-metrics and lets us show that the conformally Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of ${\lambda }_{\mathrm{ALF}}$ to positive relative mass statements for ALF metrics.

In this work, we consider a perturbation of an asymptotically conical gradient expanding Kähler-Ricci soliton metric $g$ in the same Kähler class. We demonstrate that, under suitable assumptions, the normalized Kähler-Ricci flow starting from the initial perturbed metric exists for all time and converges uniformly to an asymptotically conical gradient expanding Kähler-Ricci soliton metric $g_{\infty }$. Moreover, if the perturbed initial metric is asymptotic to $g$ at spatial infinity, then the limiting metric coincides with the original soliton, that is, $g_{\infty }=g$.

We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_Omega, mu we prove the equivalence of: (i) M_Omega, mu is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_Omega, mu and hat g_Omega, mu, and discuss open problems about the maximal domain on which the Bergman dual is defined.

We clarify the definition of polarized canonical radius in the study of Kahler Ricci flow.

We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.