Kahler Manifold Research Articles

Several injectivity theorems on compact Kähler manifolds

In this short note, we use the Bochner technique and the Hodge theory in complex differential geometry to prove several injectivity results for the cohomology of holomorphic vector bundles on compact Kahler manifolds, which generalize Enoki’s original injectivity theorem.

Complex Legendre duality

We introduce complex generalizations of the classical Legendre transform, operating on Kahler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of Kahler metrics around any real analytic Kahler metric, answering a question originating in Semmes' work.

Cohomology jump loci of quasi-compact Kähler manifolds

Cohomology jump loci of quasi-compact Kähler manifolds

Symplectic and Kähler structures on biquotients

Symplectic and Kähler structures on biquotients

A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.

Chen’s improved inequality for pointwise hemi-slant warped products in Kaehler manifolds

Recently, B.-Y. Chen discovered a technique to find the relation between second fundamental form and the warping function of warped product submanifolds. In this paper, we extend our further study of [24] by giving non-trivial examples of warped product pointwise hemi-slant submanifolds. Finally, we establish a sharp estimation for the squared norm of the second fundamental form ||h||2 in terms of the warping function f. The equality case is also investigated.

Cahen–Gutt moment map, closed Fedosov star product and structure of the automorphism group

We show that if a compact Kaehler manifold $M$ of non-negative Ricci curvature admits closed Fedosov star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cahen-Gutt moment map as in the same spirit of Calabi functional for the scalar curvature in cscK problem, and prove a Cahen-Gutt version of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kaehler manifolds. The proof uses a Hessian formula for the squared norm of Cahen-Gutt moment map.

A 3d gauge theory/quantum K-theory correspondence

The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kahler manifold X, which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the N=2 supersymmetric 3d gauge theory and the IR limit is given by Givental's permutation equivariant quantum K-theory on X. This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.

From Lagrangian to totally real geometry: coupled flows and calibrations

We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kahler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss short-time existence for our flow and show it couples well with the Streets-Tian symplectic curvature flow for almost Kahler manifolds. We also discuss possible applications to Lagrangian submanifolds and calibrated geometry.

Regularity of weak minimizers of the K-energy and applications to properness and K-stability

Let (X,ω)(X,ω) be a compact Kahler manifold and HH the space of Kahler metrics cohomologous to ωω. If a csck metric exists in HH, we show that all finite energy minimizers of the extended K-energy are smooth csck metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that the existence of a csck metric in HH implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle (X,L)(X,L) admitting a csck metric in c1(L)c1(L) is KK-polystable. When the automorphism group is finite, the properness result, combined with a result of Boucksom-Hisamoto-Jonsson, also implies that (X,L)(X,L) is uniformly K-stable.

Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kahler case. Our main question is the existence of almost Kahler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.

Some results about concircular vector fields on Riemannian manifolds

In this article, we show that the presence of a concircular vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian and Kaehler manifolds. More precisely, we find new geometrical characterizations of spheres, Euclidean spaces as well as of complex Euclidean spaces using non-trivial concircular vector fields.

Radical screen transversal slant lightlike submanfolds of indefinite Kaehler manifolds

In this paper, we define Radical screen transversal slant lightlike submanifolds of an indefinite Kaehler manifold and give an example. We prove two characterization theorems for the existence of the Radical screen transversal slant lightlike submanifolds and obtain the necessary and su_cient conditions for Radical screen transversal slant lightlike submanifolds to be Radical screen slant lightlike product.

Scalar curvature as moment map in generalized Kähler geometry

It is known that the scalar curvature arises as the moment map in Kahler geometry. In pursuit of this analogy, we introduce the notion of a moment map in generalized Kahler geometry which gives the definition of a generalized scalar curvature on a generalized Kahler manifold. From the viewpoint of the moment map, we obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. It turns out that infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite dimensional on a compact manifold. Explicit descriptions of the generalized Ricci form and the generalized scalar curvature are given on a generalized Kahler manifold of type $(0,0)$. Poisson structures constructed from a Kahler action of $T^m$ on a Kahler-Einstein manifold give intriguing deformations of generalized Kahler-Einstein structures. In particular, the anticanical divisor consists of three lines on $C P^2$ in general position yields nontrivial examples of generalized Kahler-Einsein structures

Extremal metrics on blowups along submanifolds

We give conditions under which the blowup of an extremal Kahler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo–Pacard–Singer, who considered blowups in points.

Sasakian structures on products of real line and Kahlerian manifold

In this paper, we construct a Sasakian manifold by the product of real line and K\{a}hlerian manifold with exact K\{a}hler form. This result demonstrates the close relation between Sasakian and K\{a}hlerian manifold with exact K\{a}hler form. We present an example and an open problem.

CR-slant warped product submanifolds in nearly Kaehler manifolds

We show in this paper that many well-known theorems about the geometry of warped product submanifolds of Kaehler manifolds and itself nearly Kaehler manifolds can be generalized to CR-slant warped products in nearly Kaehler manifolds.

The Concircular Curvature Tensor Of The Locally Conformal Kahler Manifold

In this research, we are calculated components conharmonic curvature tensor in some aspects Hermeation manifolding in particular of the Locally Conformal Kahler manifold. And we prove that this tensor possesses the classical symmetry properties of the Riemannian curvature. They also, establish relationships between the components of the tensor in this manifold.

Horizontally Conformal Submersions from CR-Submanifolds of Locally Conformal Kähler Manifolds

In this paper, we study horizontally conformal submersions from CR-submanifolds of a locally conformal Kahler manifold onto almost Hermitian manifolds, generalizing the results obtained by Şahin (Kodai Math. J. 31, 2008), for horizontally conformal submersions of CR-submanifolds in Kahler ambient space. In particular, we show that any horizontally homothetic submersion of a CR-submanifold M of a locally conformal Kahler manifold with Lee vector field normal to M is a Riemannian submersion up to a scale. Moreover, we obtain that such a map is harmonic, provided that the CR-submanifold is mixed geodesic.

An $$L^2$$ Dolbeault lemma and its applications to vanishing theorems

In this paper, we will first build an $$L^2$$ Dolbeault lemma by analytic methods and Hormander $$L^2$$ estimates. Then as applications, we will prove some log Nadel type vanishing theorems on compact Kahler manifolds and some log Kawamata–Viehweg type vanishing theorems on projective manifolds. Some log Nakano–Demailly type vanishing theorems for vector bundles will be also discussed by the same methods; one of which generalizes the original Nakano vanishing theorem on compact Kahler manifolds.