Narrow Results

We introduce the class of pseudo-horizontally homothetic maps from a Riemann manifold to a Kähler manifold, and we study some of their properties. For example, we prove that a pseudo-horizontally homothetic submersion is harmonic if and only if it has minimal fibres, and a pseudo-horizontally homothetic harmonic submersion pulls back complex submanifolds into minimal submanifolds.

The aim of this paper is to extend the notion of pseudo harmonic morphism introduced by Loubeau in [15] (see also [7, 4]) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic, as in [9], and pseudo-horizontally weakly conformal, see Sec. 3. We characterize them by means of germs of harmonic functions on the source polyhedron (see [13] for a precise definition) and germs of holomorphic functions on the Kähler target manifold, similarly to [15, 7].

In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds $M_n$ obtained by blowing up the diagonal of the product of two copies of a $C{P}^n$.

We show that if ${\{M_t\}}_{t\in \mathrm{\Delta }}$ is a polarized family of compact Kähler manifolds over the open unit disk $\mathrm{\Delta }$, if $N$ is a Riemannian manifold of nonpositive complexified sectional curvature, and if ${\{{\varphi }_t:M_t\to N\}}_{t\in \mathrm{\Delta }}$ is a smooth family of pluriharmonic maps, then the Dirichlet energy $E({\varphi }_t)$ is a subharmonic function of $t\in \mathrm{\Delta }$.

In this paper, we incorporate the beta constants into the defect relation and deduce a non-integrated defect relation for meromorphic maps from a Kähler manifold whose universal covering is a ball into a projective variety intersecting general divisors properly.

In each complex dimension n ≥ 2, we construct complete Kähler manifolds of bounded curvature and non-negative Ricci curvature whose Kähler–Ricci evolutions immediately acquire Ricci curvature of mixed sign.

We investigate the structure of real hypersurfaces with isometric Reeb flow in Kähler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.

We obtain curvature estimates for long-time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in [G. La Nave and G. Tian, A continuity method to construct canonical metrics, Math. Ann. 365(3) (2016) 911–921; Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218(5) (2008) 1526–1565], we adapt arguments from [F. T.-H. Fong and Y. Zhang, Local curvature estimates of long-time solutions to the Kähler–Ricci flow, Adv. Math. 375 (2020) 107416] for the Kähler–Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds.

The purpose of this paper is to construct a solution with L^p-estimates, 1≤p≤∞, to the equation on strongly q-convex domain of Kähler manifold. This is done for forms of type (n,s), s≥ max(q,k), with values in a holomorphic vector bundle which is Nakano semi-positive of type k and for forms of type (0,s), q≤s≤n-k, with values in a holomorphic vector bundle which is Nakano semi-negative of type k.

On a strictly q-convex domain D in a Kähler manifold X, we obtain the solvability of the -problem for smooth forms and currents on boundaries of D. Moreover, we study the solvability of the -problem for extensible currents.

The aim of this paper is to explore the Clairaut anti-invariant Riemannian maps from/to Kähler manifolds admitting Ricci solitons. We find the curvature relations and calculate the Ricci tensor under different conditions. We discuss the condition under which range space becomes $\alpha$-Ricci soliton. We obtain conditions for the range and kernel spaces of these maps to be Einstein. Next, we find the scalar curvature for range space. Further, we give the relation between Ricci curvature and Lie derivative under these maps. Moreover, we find the condition for a vertical potential vector field on target manifold to be conformal vector field on range space of these maps. Finally, we give non-trivial examples of such maps.

In this lecture results on the Berezin-Toeplitz quantization of arbitrary compact quantizable Kähler manifolds are presented. These results are obtained in joint work with M. Bordemann and E. Meinrenken. The existence of the Berezin-Toeplitz deformation quantization is also covered. Recent results obtained in joint work with A. Karabegov on the asymptotic expansion of the Berezin transform for arbitrary quantizable compact Kähler manifolds are explained. As an application the asymptotic expansion of the Fubini-Study fundamental form under the coherent state embedding is considered. Some comments on the dynamics of the quantum operators are given.

An algebraic variety is an object which can be defined in a purely algebraic way, starting from polynomials or more generally from finitely generated algebras over fields. When the base field is the field of complex numbers, it can also be seen as a complex manifold, and more precisely a Kähler manifold. We will review a number of notions and results related to these two aspects of complex algebraic geometry. A crucial notion is that of Hodge structure, which already appears in the Kähler context, but seems to be meaningful and interpretable only in the context of algebraic geometry.

The main goal of the paper is to make a survey and outline some new results on the existence of Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for noncompact Kähler manifolds in two important rotation invariant cases: the punctured plane (ℂ*, g*) with the complete cylindrical metric and the Kepler manifold (X, g).