Kähler-Ricci solitons on toric manifolds with positive first Chern class

In this paper we prove there exists a Kähler-Ricci soliton, unique up to holomorphic automorphisms, on any toric Kähler manifold with positive first Chern class, and the Kähler-Ricci soliton is a Kähler-Einstein metric if and only if the Futaki invariant vanishes.     

A General Inequality for Kählerian Slant Submanifolds and Related Results

An isometric immersion of a Riemannian manifold into a Kählerian manifold is called slant if it has a constant Wirtinger angle. A slant submanifold is called Kählerian slant if its canonical structure is parallel. In this article, we prove a general inequality relating the mean and scalar curvatures of Kählerian slant submanifolds in a complex space form. We also classify Kählerian slant submanifolds which satisfy the equality case of the inequality. Several related results on slant submanifolds are also proved.     

On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood

We prove that the existence of a Kähler-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 Kähler 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 Kähler potentials on which this inequality holds on higher-dimensional Fano Kähler-Einstein manifolds.     

Complex contact structures and the first eigenvalue of the dirac operator on Kähler manifolds

In this paper Kählerian Killing spinors on manifolds of complex dimensionm=4l+3 are constructed. The construction is based on a theorem which states that a closed Kähler Einstein manifold of complex dimension 4l+3 and positive scalar curvature admits a Kählerian Killing spinor if and only if there is a complex (2l+1)-contact structure. In particular, any complex contact structure in the usual sense gives rise to such a generalized contact structure. Using this, new examples of Kählerian Killing spinors are obtained.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.     

Extremal Kähler metrics on projective bundles over a curve

Let M=P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E→Σ over a compact complex curve Σ of genus ?2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.     

On the existence and nonexistence of extremal metrics on toric Kähler surfaces

In this paper we study the existence of extremal metrics on toric Kähler surfaces. We show that on every toric Kähler surface, there exists a Kähler class in which the surface admits an extremal metric of Calabi. We found a toric Kähler surface of 9 -fixed points which admits an unstable Kähler class and there is no extremal metric of Calabi in it. Moreover, we prove a characterization of the K-stability of toric surfaces by simple piecewise linear functions. As an application, we show that among all toric Kähler surfaces with 5 or 6 -fixed points, is the only one which allows vanishing Futaki invariant and admits extremal metrics of constant scalar curvature.     

A Class of Compact Kählerian Solvmanifolds and a General Conjecture

We give some explicit examples of compact Kählerian solvmanifolds and, by extending them naturally, we find a class of compact Kählerian solvmanifolds; namely, a finite quotient of a complex torus that is a holomorphic fiber bundle over a complex torus with fiber a complex torus. Then we see that under some restriction a compact solvmanifold is Kählerian if and only if it belongs to this class. We are thus led to a conjecture that this result would hold without any restriction.     

A thermodynamical formalism for Monge–Ampère equations,Moser–Trudinger inequalities and Kähler–Einstein metrics

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kähler manifold X. This functional can be seen as a generalization of Mabuchi?s K-energy functional and its twisted versions to more singular situations. Applications to Monge–Ampère equations of mean field type, twisted Kähler–Einstein metrics and Moser–Trudinger type inequalities on Kähler manifolds are given. Tian?s α-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kähler–Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Kähler metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kähler–Einstein metric, when a unique one exists, which is in line with a well-known conjecture.     

Conification construction for Kähler manifolds and its application in c-projective geometry

Two Kähler metrics on a complex manifold are called c-projectively equivalent if their J  -planar curves coincide. Such curves are defined by the property that the acceleration is complex proportional to the velocity. The degree of mobility of a Kähler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of a simply connected Kähler manifold by reducing the problem to the study of parallel Hermitian (0,2)$(0,2)$-tensors on the conification of the manifold. We also describe all such values for a Kähler–Einstein metric. We apply these results to describe all possible dimensions of the space of essential c-projective vector fields of Kähler and Kähler–Einstein metrics. We also show that two c-projectively equivalent Kähler–Einstein metrics (of arbitrary signature) on a closed manifold have constant holomorphic curvature or are affinely equivalent.     

Holomorphic Torus Actions on Compact Locally Conformal Kähler Manifolds

Given a torus action (T ², M) on a smooth manifold, the orbit map ev _x(t)=t·xfor each xMinduces a homomorphism ev _*: ²H ₁(M;). The action is said to be Rank-kif the image of ev _*has rank k(2) for each point of M. In particular, if ev _*is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold Mis Rank-1 provided that Mhas no Kähler structure.     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

Relations between the Kähler cone and the balanced cone of a Kähler manifold

In this paper, we consider a natural map from the Kähler cone of a compact Kähler manifold to its balanced cone. We study its injectivity and surjectivity. We also give an analytic characterization theorem on a nef class being Kähler.     

A Note on the Moment Map on Compact Kähler Manifolds

We consider compact Kähler manifolds acted on effectively by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||||² is constant if and only if K is semisimple and the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler manifold which is acted on trivially by K.     

Minimal Immersions of Kahler Manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einsteinor homogeneous Kähler-manifold into an Euclidean spacemust be totally geodesic. As an application, it is shown thatan open subset of the real hyperbolic plane RH² cannot be minimallyimmersed into the Euclidean space. As another application, aproof is given that if an irreducible Kähler manifold isminimally immersed in a Euclidean space, then its restrictedholonomy group must be U(n), where n = dim_CM. 2000 MathematicsSubject Classification 53B25 (primary); 53C42 (secondary).     

Symplectic reflections and complex space forms

We prove that a Hermitian manifold is a complex space form if and only if the local reflections with respect to any holomorphic surface are symplectic, i.e., preserve the Kähler form.     

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics

We propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kähler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kähler and conformal geometry such as constant scalar curvature metrics, Kähler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kähler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.     

Parabolic equations of Von Karman type on Kähler manifolds

We study a parabolic version of a system of Von Karman type on a compact Kähler manifold of arbitrary dimension. We provide local in time regular solutions, which can be extended to global bounded ones if the data of the problem are small.     

A note on the Bogomolov-type smoothness on deformations of the regular parts of isolated singularities

We apply the Tian-Todorov method, proving the Bogomolov
smoothness theorem (for deformations of compact Kähler manifolds) to deformations of the regular part of a Stein space with a finite number of isolated singular points. By the argument based on the Hodge structure on a strongly pseudo-convex Kähler domain or on a punctured Kähler space, we obtain an unobstructed subspace of the infinitesimal deformation space.