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 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.

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.

The Strong Lefschetz Property and the Schur–Weyl Duality

2013 ◽  
pp. 211-234
Author(s):  
Tadahito Harima ◽  
Toshiaki Maeno ◽  
Hideaki Morita ◽  
Yasuhide Numata ◽  
Akihito Wachi ◽  
...  
Keyword(s):  
Weyl Duality ◽  
Lefschetz Property ◽  
Strong Lefschetz Property

Cohomology Rings and the Strong Lefschetz Property

2013 ◽  
pp. 189-199
Author(s):  
Tadahito Harima ◽  
Toshiaki Maeno ◽  
Hideaki Morita ◽  
Yasuhide Numata ◽  
Akihito Wachi ◽  
...  
Keyword(s):  
Cohomology Rings ◽  
Lefschetz Property ◽  
Strong Lefschetz Property

scholarly journals The Geometry of the Weak Lefschetz Property and Level Sets of Points

2008 ◽  
Vol 60 (2) ◽  
pp. 391-411 ◽  
Author(s):  
Juan C. Migliore
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Hilbert Function ◽  
Worst Case ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Set Of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

scholarly journals GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS

2007 ◽  
Vol 04 (03) ◽  
pp. 389-436 ◽  
Author(s):  
ROGIER BOS
Keyword(s):  
Geometric Quantization ◽  
Symplectic Manifolds ◽  
Lie Algebroids ◽  
Unitary Representations ◽  
Reduction Theorem ◽  
Orbit Method ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Quantization Procedure ◽  
Hamiltonian Actions

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

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