Kähler–Einstein & K-stability

Inspired by recent work of Inoue and Hallam-Lahdili, we prove an equivariant Miyaoka-Yau inequality with a correction term for smooth Fano manifolds admitting Kähler-Ricci solitons. It recovers the usual Miyaoka-Yau inequality in the Kähler-Einstein case. Our proof uses weighted equivariant Bogomolov-Gieseker and Langer inequalities for torsion-free sheaves, together with a weighted slope estimate for the canonical extension sheaf. We also construct toric examples showing that the existence of a soliton implies neither weighted semistability of the canonical extension sheaf nor nonnegativity of the equivariant Miyaoka-Yau number.

In this paper, we prove the long time existence of the pseudo-Calabi flow on a compact toric variety for any $T^n$-invariant initial Kähler metric. If the Mabuchi functional is proper modulo automorphisms, the flow converges exponentially to a constant scalar curvature Kähler (cscK) metric. In particular, this gives a flow proof that properness of the K-energy implies the existence of a cscK metric in the toric setting.

We study the volume-normalized gradient flow of the squared Chern scalar curvature on compact Hermitian manifolds of complex dimension $n\ge 3$. The flow extends smoothly past a finite time if the spatial supremum of Chern scalar curvature belongs to $L^2$ in time. A uniform curvature bound in the evolving $L^p$ norm, for one fixed $p>n$, gives global existence and smooth convergence of the whole trajectory to a constrained critical metric. On Chern scalar-flat backgrounds, small initial $L^p$ curvature gives exponential convergence to the background. A corresponding condition on the trace-free first Chern–Ricci form gives exponential convergence on Chern–Ricci-flat Hermitian and positive Kähler–Einstein backgrounds. The smallness thresholds are independent of the initial higher derivatives. We also construct constant Chern scalar curvature backgrounds of either sign with arbitrarily small smooth perturbations whose solutions, if global, cannot converge back to the background.

In this paper, we propose the Chern-Calabi Flow for extremal metrics in conformal Hermitian geometry. After the existence of extremal metric, we consider the existence of global flows to the Chern-Calabi flows for extremal metrics on compact complex surfaces with Gauduchon metrics. To achieve this goal, we use the Adams inequality and the Gronwall inequality method to obtain fourth order derivative norm estimates. We also study the convergence of the global solution to the Chern-Calabi flow and its limit is the solution of the Chern-Calabi extremal metric problem. We propose a fourth order elliptic problem and derive the Bochner formula relevant to the elliptic estimates for the Chern-Calabi heat flow.

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 $X$.

Let $(M^{2m},g,f,J)$ be a complete nonsteady gradient Kähler-Ricci soliton satisfying $\mathrm{Ric}+{\nabla }^{2}f=\lambda g$, $\lambda \ne 0$. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as $N^{2k}\times {\mathbb{C}}^{m-k}$, where $N^{2k}$ is Kähler-Einstein with ${\mathrm{Ric}}_{g_N}=\lambda g_N$. In the shrinking case the quotient is trivial; in the normalization $\lambda =1/2$ one has $R\equiv k$. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if ${\mathcal{L}}_{\nabla f}\mathrm{Ric}$ is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching $0\le \mathrm{Ric}\le \lambda g$ forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.

We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.

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.

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.

Let $(M^{2m},g,f,J)$ be a complete gradient Kähler–Ricci shrinker satisfying $$\mathrm{Ric}+{\nabla }^{2}f=\frac{1}{2}g.$$ We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k\in \{0,\dots ,m\}$ such that $$R\equiv k$$ and $$(M^{2m},g,J)\cong (N^{2k}\times {\mathbb{C}}^{m-k},g_N+g_{\mathrm{Euc}},J_N\oplus J_0),$$ where $(N^{2k},g_N,J_N)$ is Kähler–Einstein and $${\mathrm{Ric}}_{g_N}=\frac{1}{2}{g}_N.$$ As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition ${\mathcal{L}}_{\nabla f}\mathrm{Ric}=0$ implies radial flatness and hence rigidity.

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 that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies $\rho =C\omega +nd{d}^{c}f$, and which satisfy $C>0$. With analogous assumptions on the flow, we obtain subconvergence to an $f$-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that $(M,g,f)$ is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full 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.

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 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 prove that the asymptotics of the Hermitian-Yang-Mills flow on a slope semistable holomorphic vector bundle over a compact Kähler manifold are determined algebro-geometrically, via the iterated filtration defined by Haiden-Katzarkov-Kontsevich-Pandit. This proves a conjecture of Haiden-Katzarkov-Kontsevich-Pandit in this setting. Moreover, we prove a non-linear analogue, relating the asymptotics of the Calabi flow near a cscK manifold to the iterated balancing filtration of the deformation space. In both settings, we reduce the infinite-dimensional flow to a finite-dimensional flow, following the foundational work of Chen-Sun. In finite dimensions, we prove that the asymptotics of the moment map flow are determined by the iterated balancing filtration, proving a conjecture of Ibáñez Núñez.

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 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.