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.     

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 .     

Sectional curvatures of Kähler moduli

If X is a compact Kähler manifold of dimension n, we let denote the cone of Kähler classes, and the level set given by classes D with Dⁿ=1. This space is naturally a Riemannian manifold and is isometric to the manifold of Kähler forms with ⁿ some fixed volume form, equipped with the Hodge metric, as studied previously by Huybrechts. We study these spaces further, in particular their geodesics and sectional curvatures. Conjecturally, at least for Calabi–Yau manifolds and probably rather more generally, these sectional curvatures should be bounded between and zero. We find simple formulae for the sectional curvatures, and prove both the bounds hold for various classes of varieties, developing along the way a mirror to the Weil–Petersson theory of complex moduli. In the case of threefolds with h^1,1=3, we produce an explicit formula for this curvature in terms of the invariants of the cubic form. This enables us to check the bounds by computer for a wide range of examples. Finally, we explore the implications of the non-positivity of these curvatures.     

Uniqueness of extremal Kähler metrics

In the infinite dimensional space of Kähler potentials, the geodesic equation of disc type is a complex homogenous Monge–Ampère equation. The partial regularity theory established by Chen and Tian [C. R. Acad. Sci. Paris, Ser. I 340 (5) (2005)] amounts to an improvement of the regularity of the known $C^{1\text{,}1}$ solution to the geodesic of disc type to almost everywhere smooth. For such an almost smooth solution, we prove that the K-energy functional is sub-harmonic along such a solution. We use this to prove the uniqueness of extremal Kähler metrics and to establish a lower bound for the modified K-energy if the underlying Kähler class admits an extremal Kähler metric. To cite this article: X.X. Chen, G. Tian, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Equivariant Kähler compactifications of homogeneous manifolds

A compact complex manifoldX is an equivariant compactification of a homogeneous manifoldG/H (G a connected complex Lie group,H a closed complex subgroup ofG), if there exists a holomorphic action ofG onX such that theG-orbit of some pointx inX is open and H is the isotropy group ofx. GivenG andH, for some groups (e.g.,G nilpotent) there are necessary and sufficient conditions for the existence of an equivariant Kähler compactification which are proven in this paper.     

Bounding Diameter of Conical Kähler Metric

In this paper we research the differential geometric and algebro-geometric properties of the noncollapsing limit in the conical continuity equation which generalize the theory in La Nave et al. in Bounding diameter of singular Kähler metric, arXiv:1503.03159v1 [23].     

Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal.     

On quasi-Sasakian hypersurfaces of Kähler manifolds

We establish several conditions which are necessary for a quasi-Sasakian hypersurface of a Kähler manifold to be minimal.     

Quaternionic Kähler reductions of Wolf spaces

The main purpose of the following article is to introduce a Lie theoretical approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called generalized Wolf spaces.     

Light-like CR hypersurfaces of indefinite Kähler manifolds

We study a new class of real hypersurfaces called Light-like CR hypersurfaces, of indefinite Kahler manifolds, and claim several new results of geometrical/physical significance. In particular, we show that our study has a direct relation with the physically important asymptotically flat spacetimes; which further lead to the Twistor theory of Penrose and the Heaven theory of Newman. As the induced connection, on the degenerate hypersurface, may not be a metric connection, we overcome this difficulty by using differential geometric technique and deduce the embedding conditions called Gauss-Codazzi equations. Finally, we find the integrability conditions for all the possible distributions and specialize the embedding conditions when the ambient space is a complex space form. We add to the list of totally umbilical nondegenerate hypersurfaces [16] the totally umbilical light-like cone, in the degenerate case, and prove the nonexistence of totally umbilical light-like CR hypersurfaces in ¯M(c) withc 0 (see Yano and Kon [22] and Tashiro and Tachibana [20] for the nondegenerate case).     

Positive Twistor Bundle of a Kähler Surface

The aim of this paper is to characterize Kähler surfaces in terms oftheir positive twistor bundle. We prove that an oriented four-dimensionalRiemannian manifold (M, g) admits a complex structure J compatible with the orientation and such that (M, g, J is a Kähler manifold ifand only if the positive twistor bundle (Z ⁺(M), g _c ) admits a verticalKilling vector field.     

On a class of compact Kähler varieties

Summary Is given in the introduction.

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Sunto Si studiauo le varietà k?hleriane compatte, V, dotate di proiezioni olomorfe sopra spazi proiettivi complessi di dimensione inferiore. Si dimostra, fra l'altro, che, qualora tali proiezioni olomorfe soddisfino ad opportune condizioni di regolarità, le restrizioni di certi gruppi diDolbeault di V alle fibre (non singolari) determinate dalle proiezioni olomorfe suddette, sono iniettive. Da ciò consegue che il genere geometrico ed altri invarianti di V si annullano.

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This paper was supported by the National Science Foundation, under contract NSF G-4143 with the Northwestern University. Some of the results of the present paper have been announced, without proof, inE. Vesentini,Sopra i sistemi fibrati k?hleriani compatti, ? Rend. Acc. Naz. Lincei ?, (8)24 (1958).     

Examples of asymptotically conical Ricci-flat Kähler manifolds

Previously the author has proved 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 ${H^2_c(Y,\mathbb R)}$ . These manifolds can be considered to be generalizations of the Ricci-flat ALE Kähler spaces known by the work of P. Kronheimer, D. Joyce and others. This article considers further the problem of constructing examples. We show that every 3-dimensional Gorenstein toric Kähler cone admits a crepant resolution for which the above theorem applies. This gives infinitely many examples of asymptotically conical Ricci-flat manifolds. Then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit Ricci-flat Kähler cone metrics by the work of C. Boyer, K. Galicki, J. Kollár, and others. We concentrate on 3-dimensional examples. Two families of hypersurface examples are given which are distinguished by the condition b ₃(Y) = 0 or b ₃(Y) ≠ 0.     

Real Lightlike Hypersurfaces of Paraquaternionic Kähler Manifolds

The main purpose of this paper is to give basic properties of real lightlike hypersurfaces of paraquaternionic manifold and to prove the nonexistence of real lightlike hypersurfaces in paraquaternionic space forms under some conditions. Dedicated to the memory of Professor Aldo Cossu     

Hermitian matrices and cohomology of Kähler varieties

We give some upper bounds on the dimension of the kernel of the cup product map $H^{1}(X,\mathbb{C}) \otimes H^{1}(X,\mathbb{C}) \to H^{2}(X,\mathbb{C})We give some upper bounds on the dimension of the kernel of the cup product map , where X is a compact K?hler variety without Albanese fibrations.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.     

Nearly Kähler Submanifolds of a Space Form

In this article, we study isometric immersions of nearly Kähler manifolds into a space form (especially Euclidean space) and show that every nearly Kähler submanifold of a space form has an umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover, using this foliation we show that there is no non-homogeneous 6-dimensional nearly Kähler submanifold of a space form. We prove some results towards a classification of nearly Kähler hypersurfaces in standard space forms.     

Dynamics of automorphisms on compact Kähler manifolds

We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.