On the formality and strong Lefschetz property of symplectic manifolds

2008 ◽  
Vol 145 (2) ◽  
pp. 363-377 ◽  
Author(s):  
TAEK GYU HWANG ◽  
JIN HONG KIM
Keyword(s):  
Symplectic Manifold ◽  
Explicit Construction ◽  
Symplectic Manifolds ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Fixed Point Set ◽  
Lefschetz Property ◽  
The Difference ◽  
Simply Connected ◽  
Strong Lefschetz Property

AbstractThe main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kähler and symplectic manifolds. To be precise, it is known that simply connected symplectic manifolds of dimension greater than 8, in general, do not satisfy the formality satisfied by all Kähler manifolds. In this paper we show that such non-formality of simply connected symplectic manifolds occurs even in dimension 8. We do this by some complicated but explicit construction of a simply connected non-formal symplectic manifold of dimension 8. In this construction we essentially use a variation of the construction of a simply connected symplectic manifold by Gompf. As a consequence, we can give infinitely many simply connected non-formal symplectic manifolds of any even dimension no less than 8.Secondly, we show that every compact symplectic manifold admitting a semi-free Hamiltonian circle action with only isolated fixed points must satisfy the strong Lefschetz property satisfied by all Kähler manifolds. This result shows that the strong Lefschetz property for the symplectic manifold admitting Hamiltonian circle actions is closely related to their fixed point set, as expected.

scholarly journals On the formality and strong Lefschetz property of symplectic manifolds – Erratum

2009 ◽  
Vol 147 (1) ◽  
pp. 255-255
Author(s):  
Taek Kyu Hwang ◽  
Jin Hong Kim
Keyword(s):  
Fundamental Group ◽  
Symplectic Manifold ◽  
Symplectic Manifolds ◽  
Finite Covering ◽  
Massey Product ◽  
Lefschetz Property ◽  
The Stability ◽  
Simply Connected ◽  
Open Question ◽  
Strong Lefschetz Property

Professor Vicente Muñoz kindly informed us that there is an inaccuracy in Lemma 3.5 of [1]. The correct statement of Lemma 3.5 is now that the fundamental group π1(X′) of the manifold X′ is Z, since the monodromy coming from φ8 does not imply that g4 = g4−1. Therefore, what we have actually constructed in Section 3 of [1] is a closed non-formal 8-dimensional symplectic manifold with π1 = Z whose triple Massey product is non-zero, so that the simply-connectedness in Theorem 1.1 should be dropped. As far as we know, the existence of a simply connected closed non-formal 8-dimensional symplectic manifold whose triple Massey product is non-zero still remains an open question. All other main results, especially Theorem 1.2 and Corollary 1.3, in [1] are not affected by this mistake. Furthermore, the stability of the non-formality under a finite covering as in Subsection 3.3 holds in general. We want to thank Professor Muñoz for his careful reading.

Coupled Vortex Equations on Complete Kähler Manifolds

2008 ◽  
Vol 51 (3) ◽  
pp. 467-480
Author(s):  
Yue Wang
Keyword(s):  
Symmetric Spaces ◽  
Existence Results ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Convex Domains ◽  
Hermitian Symmetric Spaces ◽  
Vortex Equations ◽  
Curved Manifolds ◽  
Pseudo Convex ◽  
Simply Connected

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

scholarly journals Proof of the radical conjecture for homogeneous Kähler manifolds

1989 ◽  
Vol 114 ◽  
pp. 77-122 ◽  
Author(s):  
Josef Dorfmeister
Keyword(s):  
Lie Algebra ◽  
Kähler Manifold ◽  
Transitive Group ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Fiber Space ◽  
Kahler Manifold ◽  
Solvable Ideal ◽  
Simply Connected ◽  
Bounded Homogeneous Domain

In 1967 Gindikin and Vinberg stated the Fundamental Conjecture for homogeneous Kähler manifolds. It (roughly) states that every homogeneous Kähler manifold is a fiber space over a bounded homogeneous domain for which the fibers are a product of a flat with a simply connected compact homogeneous Kähler manifold. This conjecture has been proven in a number of cases (see [6] for a recent survey). In particular, it holds if the homogeneous Kähler manifold admits a reductive or an arbitrary solvable transitive group of automorphisms [5]. It is thus tempting to think about the general case. It is natural to expect that lack of knowledge about the radical of a transitive group G of automorphisms of a homogeneous Kähler manifold M is the main obstruction to a proof of the Fundamental Conjecture for M. Thus it is of importance to consider the Kähler algebra generated by the radical of the Lie algebra of G. Computations in this context suggest that one rather considers Kähler algebras generated by an arbitrary solvable ideal.

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}}.

scholarly journals Canonical equivariant extensions using classical Hodge theory

2005 ◽  
Vol 2005 (8) ◽  
pp. 1277-1282 ◽  
Author(s):  
Christopher Allday
Keyword(s):  
Hodge Theory ◽  
Symplectic Manifolds ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Hamiltonian Actions

Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory.

scholarly journals Formality and the Lefschetz property in symplectic and cosymplectic geometry

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Giovanni Bazzoni ◽  
Marisa Fernández ◽  
Vicente Muñoz
Keyword(s):  
Betti Numbers ◽  
Symplectic Manifolds ◽  
Topological Properties ◽  
Cosymplectic Manifolds ◽  
Connected Case ◽  
Lefschetz Property ◽  
Simply Connected

AbstractWe review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b

scholarly journals Automorphisms and examples of compact non-Kähler manifolds

2017 ◽  
Vol 121 (1) ◽  
pp. 49
Author(s):  
Gunnar Þór Magnússon
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Canonical Bundle ◽  
Complex Manifolds ◽  
Kahler Manifold ◽  
Simply Connected

If $f$ is an automorphism of a compact simply connected Kähler manifold with trivial canonical bundle that fixes a Kähler class, then the order of $f$ is finite. We apply this well known result to construct compact non-Kähler manifolds. These manifolds contradict the abundance and Iitaka conjectures for complex manifolds.

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.

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