Yau conjectured that a Fano manifold admits a Kahler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian, Donaldson and others. The Mabuchi energy functional plays a central role in these ideas.
We study the E_k functionals introduced by X.X. Chen and G. Tian which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kahler-Einstein metric then the functional E_1 is bounded from below, and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen. We show in fact that E_1 is proper if and only if there exists a Kahler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability.
We also show that on a Fano Kahler-Einstein manifold all of the functionals E_k are bounded below on the space of metrics with nonnegative Ricci curvature.
In this paper we study the extension of holomorphic canonical forms on complete d-bounded Kahler manifolds by using L2 analytic methods and L2 Hogde theory, which generalizes some classical results to noncompact cases.
In this paper we consider left-invariant pseudo-Kähler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated pseudo-Kähler metrics are investigated. It is proved that the associated pseudo-Kähler metric is Ricci-flat, that the curvature tensor has zero pseudo-Riemannian norm, and that the curvature tensor has some non-zero components that depend only on two or, at most, three parameters. The pseudo-Kähler structures obtained give basic models of pseudo-Kähler six-dimensio…
The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost Kähler manifold, and show that in the previous example (and in another one obtained from a Kodaira surface) the Ricci curvature of the almost Kähler metric vanishes for all …
In this short note we are concerned with the Kahler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kahler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold, which provides a local model for the future study of the global Kahler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is a desired one. In the second approach, we obtain a complet…
We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on $(m,0)$ forms, i.e., sections of the canonical bundle of Kähler manifolds, where $m$ is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of $(m, 0)$ forms, which involves only the Ricci curvature and the gradient of the scalar curvature.
This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical Kähler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that…
For projective varieties with definite first Chern class we have one type of canonical metric which is called Kähler-Einstein metric. But for varieties with an intermidiate Kodaira dimension we can have several different types of canonical metrics. In this paper we introduce a new notion of canonical metric for varieties with an intermidiate Kodaira dimension. We highlight that to get $C^\infty$-solution of CMA equation of relative Kähler Einstein metric we need Song-Tian-Tsuji measure (which has minimal singularities with respect to other relative volume forms) be $C^\infty$-smooth and specia…
In this paper, we consider Kahler-Ricci flow on n-dimensional Kahler manifold with semi-ample canonical line bundle and 0< m:= Kod(X)<n. Such manifolds admit a Calabi-Yau fibration over its canonical model. We prove that the scalar curvature of the Kahler metric along the normalized Kahler-Ricci flow converge to -m outside the singular set of this fibration.
In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the canonical measure of the LC pair. We prove that the relative canonical measure on a projective family of LC pairs of log general type defines a singular hermitian metric on the relative log canonical bundle and the metric has semipositive curvature in the sense of current. This is…
It is a classical fact that the cotangent bundle $T^* \M$ of a differentiable manifold $\M$ enjoys a canonical symplectic form $Ω^*$. If $(\M,\j,g,ω)$ is a pseudo-Kähler or para-Kähler $2n$-dimensional manifold, we prove that the tangent bundle $T\M$ also enjoys a natural pseudo-Kähler or para-Kähler structure $(\J,\G,Ω)$, where $Ω$ is the pull-back by $g$ of $Ω^*$ and $\G$ is a pseudo-Riemannian metric with neutral signature $(2n,2n)$. We investigate the curvature properties of the pair $(\J,\G)$ and prove that: $\G$ is scalar-flat, is not Einstein unless $g$ is flat, has nonpositive (resp.\ …
In this paper we prove that the Kähler-Einstein metrics for a toroidal canonical degeneration family of Kähler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also prove the incompleteness of the Weil-Peterson metric in this case.
We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings.
In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni concerning manifolds with negative $k$-Ricci curvature and generalize a result of Wu-Yau and Diverio-Trapa…
In this paper, we generalize Chen-Tian energy functionals to Kähler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of Kähler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu's holomorphic invariant.
On a Fano manifold, we prove that the Kahler-Ricci flow starting from a Kahler metric in the anti-canonical class which is sufficiently close to a Kahler-Einstein metric must converge in a polynomial rate to a Kahler-Einstein metric. The convergence can not happen in general if we study the flow on the level of Kahler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's μfunctional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main te…
We survey some recent developments in the study of canonical Kähler metrics on algebraic varieties and their relation with stability in algebraic geometry.
In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the $L^2$-norm of the scalar curvature along the Kähler-Ricci flow.
We consider complete nearly Kähler manifolds with the canonical Hermitian connection. We prove some metric properties of strict nearly Kähler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly Kähler manifold to be a twistor space over a quaternionic Kähler manifold is given. This enables us to give classification results in 10-dimensions.
The existence of Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the Kähler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized Kähler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metri…
We construct Kahler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable. In complex dimension 2, we show that any complete Kahler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to one of the model metrics constructed by Kobayashi and Nakamura arising from hyperbolic geometry.
We prove that a Kahler metric in the anticanonical class which is a critical point of the functional E_k and has nonnegative Ricci curvature, is necessarily Kahler-Einstein. This partially answers a question of X.X.Chen.
We consider the Kähler Ricci flow on a smooth minimal model of general type, we show that if the Ricci curvature is uniformly bounded below along the Kähler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the flow is homeomorphic to the canonical model. Moreover, we can give a purely analytic proof of a recent result of Tosatti-Zhang (\cite{TZ}) that if the canonical line bundle $K_X$ is big and nef, but not ample, then the flow is of Type IIb.
We numerically calculate Perelman's entropy for a variety of canonical metrics on $\mathbb{CP}^{1}$-bundles over products of Fano Kähler-Einstein manifolds. The metrics investigated are Einstein metrics, Kähler-Ricci solitons and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question.
We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge Kähler class a conformally Kähler, Einstein--Maxwell metric, or more generally, a Kähler metric of constant $(ξ, a, p)$-scalar curvature, then this metric minimizes the $(ξ,a,p)$-Mabuchi functional. Our method of proof extends the approach introduced by Donaldson and developed by Li and Sano--Tipler, via finite dimensional approximations and generalized balanced metrics. As an application of our result and the recent construction of Koca--Tønnesen-Friedman, we describe the Kähler classes on a geometr…
In the compact setting, Aazami and Ream \cite{Aazami:2022th} proved that Riemannian metrics dual to a class of Lorentzian metrics, called (compact) general plane-fronted waves, are almost-Kähler. In this note, we explain how to construct extremal and second-Chern-Einstein non-Kähler almost-Kähler metrics dual to those general plane-fronted waves.
We consider constant scalar curvature Kähler metrics on a smooth minimal model of general type in a neighborhood of the canonical class, which is the perturbation of the canonical class by a fixed Kähler metric. We show that sequences of such metrics converge smoothly on compact subsets away from a subvariety to the singular Kähler Einstein metric in the canonical class. This confirms partially a conjecture of Jian-Shi-Song about the convergence behavior of such sequences.
In this paper, we show that along $\mathbb Q$-Fano fibration, when general fibres, base and central fiber (with at worst Kawamata log terminal singularities)are K-poly stable then there exists a relative Kähler-Einstein metric. We introduce the fiberwise Kähler-Einstein foliation and we mention that the main difficulty to obtain higher estimates is to solve relative CMA equation along such foliation. We propose a program such that for finding a pair of canonical metric $(ω_X,ω_B)$, which satisfies in $$Ric(ω_X)=π^*ω_B+π^*(ω_{WP})+[\mathcal N]$$ on K-poly stable degeneration $π:X\to B$, where $…
We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature, which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler C-space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of positive scalar curvature
We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existence of coupled Kähler-Einstein metrics imply a certain algebraic stability condition, generalizing $K$-polystability.
In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an example where the modified K energy is bounded from below but not proper.
On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.
We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S^2-bundle over H^4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we …
An explicit seminorm $||f||_{#}$ on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to reproduce the Mabuchi energy functional of the corresponding projective variety. Thus the boundedness from below of the Mabuchi functional, and hence the existence of Kähler-Einstein metrics, is related to the behavior of the current $[Y_s]$ and the seminorm $||f||_{#}$ along the or…
We prove that the existence of a Kahler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kahler metrics with positive Ricci curvature. We also prove that these energy functionals are bounded from below on this set if and only if one of them is. This answers two questions raised by X.-X. Chen. As an application, we obtain a new proof of the classical Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a canonical enlargement of the space of Kahler potentials on which thi…
We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds of complex dimension three, and for projective manifolds of all dimensions if assuming the abundance conjecture.
We study a sequence of connections which is associated with a Riemannian metric and an almost symplectic structure on a manifold. We prove that if this sequence is trivial (i.e. constant) or 2-periodic, then the manifold has a canonical Kähler structure.
We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal Kähler metrics on a compact Kähler manifold introduced in our previous work. This extends a result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal Kähler case. Furthermore, we show that a weighted extremal Kähler metric is a global minimum of a suitable weighted version of the modified Mabuchi energy. This implies a suitable notion of weighted K-semistability of a Kähler manifold admitting a weighted extremal Kähler metric.
In this paper, we prove that the Kahler Ricci flow converges to a Kahler Einstein metric when E_1 energy is small. We also prove that E_1 is bounded from below if and only if the K energy is bounded from below in the canonical class.
In this paper, we study the boundary behavior of the negatively curved Kähler-Einstein metric attached to a log canonical pair $(X,D)$ such that $K_X+D$ is ample. In the case where $X$ is smooth and $D$ has simple normal crossings support (but possibly negative coefficients), we provide a very precise estimate on the potential of the KE metric near the boundary $D$. In the more general singular case ($D$ being assumed effective though), we show that the KE metric has mixed cone and cusp singularities near $D$ on the snc locus of the pair. As a corollary, we derive the behavior in codimension o…
Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the finite energy space $\mathcal E^1_ω(X)$. Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of Kähler-Einstein metrics on log Fano pairs is …
In this paper we prove that the Kähler-Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil-Peterson metric in this case.
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of $X$ over its canonical model $X_{can}$. This assumption on the Ricci curvature always holds when the Kodaira dimension of $X$ is $n$, $n-1$ or when the general fibre of $X$ over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to…
Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a Kähler-Einstein metric iff $X$ has semi-log canonical singularities i.e. iff $X$ is a stable variety in the sense of Kollár-Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kähler-Einstein metric in this singular context simply means a Kähler-Einstein on the regular locus of $X$ with volume equal to the algebraic volume…
In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the Kähler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functionals in case of Kaehler manifolds. On Kähler-Einstein manifold with positive scalar curvature, if the initial metric has positive bisectional curvature, we prove that these functionals have a uniform lower bound, via the effective use of Tian's inequality. Consequently, we prove …
Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a Kähler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature Kähler metrics in any rational Kähler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in Kähler classes that are close to the pullback of the Fubini--Study class.
In the following article we study the limiting properties of the Yang-Mills flow associated to a holomorphic vector bundle E over an arbitrary compact Kähler manifold (X,ω). In particular we show that the flow is determined at infinity by the holomorphic structure of E. Namely, if we fix an integrable unitary reference connection A_0 defining the holomorphic structure, then the Yang-Mills flow with initial condition A_0, converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E_{\infty}, which is isomorphic to…
