A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in Cn

Let $(M,g)$ be a simply connected complete Kähler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in ${\mathbb{C}}^n$, where ${\dim }_{\mathbb{C}}M=n$. To cite this article: H. Seshadri, K. Verma, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Interpolation in non-positively curved Kähler manifolds

We extend to any complete simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in\(\mathbb{C}\) and in the unit disk given by Berndtsson, Ortega-Cerdà and Seip. The main tools are L² estimates and a comparison theorem for the Hessian in Kähler geometry due to Greene, Wu, and Siu, Yau.     

Symmetric differentials, Kähler groups and ball quotients

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients:

• Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kähler manifold X are controlled by the global symmetric differentials on X.
• An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

     

A Kähler structure of the triplectic geometry

We study the geometry of the triplectic quantization of gauge theories. We show that the triplectic geometry is determined by the geometry of a Kähler manifoldN endowed with a pair of transversal polarizations. The antibrackets can be brought to the canonical form if and only ifN admits a flat symmetric connection that is compatible with the complex structure and the polarizations.     

A note on almost Kähler and nearly Kähler submersions

In this paper the integrability of the horizontal distribution of an almost-Kähler or a nearly-Kähler submersion is studied and curvature properties of such submersions are investigated.     

A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold

A class of minimal almost complex submanifolds of a Riemannian manifold with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifold of non zero scalar curvature, in particular, when is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M²ⁿ of maximal possible dimension 2n. More precisely, we prove that any such Kähler submanifold M²ⁿ of is the projection of a holomorphic Legendrian submanifold of the twistor space of , considered as a complex contact manifold with the natural holomorphic contact structure . Any Legendrian submanifold of the twistor space is defined by a generating holomorphic function. This is a natural generalization of Bryants construction of superminimal surfaces in S⁴=P¹. Mathematics Subject Classification (1991) Primary: 53C40; Secondary: 53C55     

A Gradient Estimate for the Ricci–Kähler Flow

We show that for a complete solution to theRicci–Kähler flow where the curvature, the potential andscalar curvature functions and their gradients are bounded depending ontime, the absolute value of both the scalar curvature and the gradientsquared of a modified potential function are bounded byC/t.     

Kähler manifolds with homothetic foliation by curves

The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliations by curves.     

A modified Kähler–Ricci flow

In this note, we study a Kähler–Ricci flow modified from the classic version. In the non-degenerate case, strong convergence at infinite time is achieved. The main focus should be on degenerate case, where some partial results are presented.     

A uniqueness theorem in Kähler geometry

We consider compact Kähler manifolds with their Kähler Ricci tensor satisfying F(Ric) = constant. Under the nonnegative bisectional curvature assumption and certain conditions on F, we prove that such metrics are in fact Kähler–Einstein.     

Kähler submanifolds of the symmetrized polydisc

This paper proves the non-existence of common Kähler submanifolds of the complex Euclidean space and of the symmetrized polydisc endowed with their canonical metrics.     

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.     

Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in ${H^2_c(Y,\mathbb{R})}We prove that a crepant resolution π : Y → X of a Ricci-flat K?hler cone X admits a complete Ricci-flat K?hler metric asymptotic to the cone metric in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A K?hler cone (X,[`(g)]){(X,\bar{g})} is a metric cone over a Sasaki manifold (S, g), i.e. ${X=C(S):=S\times\mathbb{R}_{ >0 }}${X=C(S):=S\times\mathbb{R}_{ >0 }} with [`(g)]=dr² +r² g{\bar{g}=dr^2 +r^2 g}, and (X,[`(g)]){(X,\bar{g})} is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat K?hler metrics on crepant resolutions p:Y? X=\mathbbC<sup>n</sup> /G{\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}, with G ì SL(n,\mathbbC){\Gamma\subset SL(n,\mathbb{C})}, due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric K?hler cone admits a Ricci-flat K?hler cone metric. It follows that if a toric K?hler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T <sup>n</sup> -invariant Ricci-flat K?hler metric asymptotic to the cone metric (X,[`(g)]){(X,\bar{g})} in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry.     

Kähler magnetic fields on Kähler manifolds of negative curvature

On a Kähler manifold we have natural uniform magnetic fields which are constant multiples of the Kähler form. Trajectories, which are motions of electric charged particles, under these magnetic fields can be considered as generalizations of geodesics. We give an overview on a study of Kähler magnetic fields and show some similarities between trajectories and geodesics on Kähler manifolds of negative curvature.     

Non-positively curved 3-manifolds with non-Kähler π1

In this Note a simple rank estimate for harmonic maps, with domain a Kähler manifold and target a compact 3-manifold, is proved. As an application we give new examples of fundamental groups of compact manifolds which are not Kähler groups.     

Regularity of the Kähler–Ricci flow

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.     

A Scalar Curvature Bound Along the Conical Kähler–Ricci Flow

Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler–Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.     

Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves

In a given Kähler manifold (M,J) we introduce the notion of Kähler Frenet curves, which is closely related to the complex structure J of M. Using the notion of such curves, we characterize totally geodesic Kähler immersions of M into an ambient Kähler manifold and totally geodesic immersions of M into an ambient real space form of constant sectional curvature .