scholarly journals Vanishing and estimation results for Hodge numbers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Peter Petersen ◽  
Matthias Wink
Keyword(s):  
Cohomology Ring ◽  
Kähler Manifolds ◽  
Curvature Operator ◽  
Kahler Manifolds ◽  
Weighted Sum ◽  
Rational Cohomology ◽  
Hodge Numbers ◽  
The Individual ◽  
Complex Projective ◽  
Estimation Theorem

Abstract We show that compact Kähler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the Kähler curvature operator is positive. This follows from a more general vanishing and estimation theorem for the individual Hodge numbers. We also prove an analogue of Tachibana’s theorem for Kähler manifolds.

scholarly journals The Hodge ring of Kähler manifolds

2013 ◽  
Vol 149 (4) ◽  
pp. 637-657 ◽  
Author(s):  
D. Kotschick ◽  
S. Schreieder
Keyword(s):  
Complete Solution ◽  
Classical Problem ◽  
Kähler Manifolds ◽  
Topological Invariants ◽  
Kahler Manifolds ◽  
Linear Combinations ◽  
Projective Varieties ◽  
Chern Numbers ◽  
Hodge Numbers ◽  
Complex Projective

AbstractWe determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kähler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kähler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch’s.

scholarly journals Strongly not relatives Kähler manifolds

2017 ◽  
Vol 4 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Michela Zedda
Keyword(s):  
Projective Space ◽  
Positive Constant ◽  
Complex Projective Space ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Infinite Dimensional ◽  
Kahler Manifold ◽  
Hartogs Domains ◽  
Complex Projective

Abstract In this paper we study Kähler manifolds that are strongly not relative to any projective Kähler manifold, i.e. those Kähler manifolds that do not share a Kähler submanifold with any projective Kähler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full Kähler immersion into the infinite dimensional complex projective space. As application we get that the 1-parameter families of Bergman-Hartogs and Fock-Bargmann-Hartogs domains are strongly not relative to projective Kähler manifolds.

scholarly journals Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

2014 ◽  
Vol 151 (2) ◽  
pp. 351-376 ◽  
Author(s):  
Fréderic Campana ◽  
Benoît Claudon ◽  
Philippe Eyssidieux
Keyword(s):  
General Type ◽  
Linear Representation ◽  
Kähler Manifolds ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Shafarevich Conjecture ◽  
Classical Work ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Holomorphic Convexity

AbstractWe extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.

scholarly journals The canonical spectrum of projective toric manifolds

2018 ◽  
Vol 29 (07) ◽  
pp. 1850048
Author(s):  
Mounir hajli
Keyword(s):  
Spectral Theory ◽  
Spectral Properties ◽  
Kähler Manifolds ◽  
Combinatorial Structure ◽  
Kahler Manifolds ◽  
Toric Manifold ◽  
Closed Formula ◽  
Toric Manifolds ◽  
Complex Projective

Let [Formula: see text] be a complex projective toric manifold. We associated to [Formula: see text], a positive and closed [Formula: see text]-current called the canonical toric Kähler current of [Formula: see text] denoted by [Formula: see text], and a new invariant called the canonical spectrum of [Formula: see text]. This spectrum is obtained as the set of the eigenvalues of a singular Laplacian defined by [Formula: see text] and which is described uniquely by the combinatorial structure of [Formula: see text]. The construction of this Laplacian and the study of its spectral properties are the consequence of a generalized spectral theory of Laplacians on compact Kähler manifolds that we develop in this paper.

Almost isotropic Kähler manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 1-16
Author(s):  
Benjamin Schmidt ◽  
Krishnan Shankar ◽  
Ralf Spatzier
Keyword(s):  
Classical Result ◽  
Jacobi Operator ◽  
Unit Vector ◽  
Complete Riemannian Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Jacobi Operators ◽  
Sectional Curvatures ◽  
Simply Connected ◽  
Complex Projective

AbstractLet M be a complete Riemannian manifold and suppose {p\in M}. For each unit vector {v\in T_{p}M}, the Jacobi operator, {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v}. Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M}. If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds, i.e. manifolds having the property that for each {p\in M}, there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M}. Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}}.

Kähler groups and rigidity phenomena

1991 ◽  
Vol 109 (1) ◽  
pp. 31-44 ◽  
Author(s):  
F. E. A. Johnson ◽  
E. G. Rees
Keyword(s):  
Kähler Manifolds ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Projective Varieties ◽  
Group Extensions ◽  
Finitely Presented Groups ◽  
Kähler Groups ◽  
Complex Projective ◽  
Finitely Presented

The class of fundamental groups of non-singular complex projective varieties is an interesting, but as yet imperfectly understood, class of finitely presented groups. Membership of is known to be extremely restricted (see [22, 23]). In this paper, we employ geometrical rigidity properties to realize some group extensions as elements of as in our previous papers, we find it convenient to work simultaneously with the class ℋ of fundamental groups of compact Kähler manifolds.

scholarly journals REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS

2013 ◽  
Vol 24 (07) ◽  
pp. 1350050 ◽  
Author(s):  
JURGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Kähler Manifolds ◽  
Real Hypersurfaces ◽  
Kahler Manifolds ◽  
Totally Geodesic ◽  
Open Part ◽  
Complex Submanifold ◽  
Complex Projective

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.

Topology of Some Kähler Manifolds II

1969 ◽  
Vol 12 (4) ◽  
pp. 457-460 ◽  
Author(s):  
K. Srinivasacharyulu
Keyword(s):  
Euler Number ◽  
Complex Surface ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Almost Complex ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Simply Connected ◽  
Complex Projective ◽  
Positively Curved

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1 (C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

Sign in / Sign up

Export Citation Format

Share Document