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On ( 1 , 1 ) -tensor fields on symplectic manifolds

Anton Dekrét (1999)

Archivum Mathematicum

Two symplectic structures on a manifold M determine a (1,1)-tensor field on M . In this paper we study some properties of this field. Conversely, if A is (1,1)-tensor field on a symplectic manifold ( M , ω ) then using the natural lift theory we find conditions under which ω A , ω A ( X , Y ) = ω ( A X , Y ) , is symplectic.

On a certain generalization of spherical twists

Yukinobu Toda (2007)

Bulletin de la Société Mathématique de France

This note gives a generalization of spherical twists, and describe the autoequivalences associated to certain non-spherical objects. Typically these are obtained by deforming the structure sheaves of ( 0 , - 2 ) -curves on threefolds, or deforming ℙ -objects introduced by D.Huybrechts and R.Thomas.

On a generalized Calabi-Yau equation

Hongyu Wang, Peng Zhu (2010)

Annales de l’institut Fourier

Dealing with the generalized Calabi-Yau equation proposed by Gromov on closed almost-Kähler manifolds, we extend to arbitrary dimension a non-existence result proved in complex dimension 2 .

On a volume element of a Hitchin component

Yaşar Sözen (2012)

Fundamenta Mathematicae

Let Σ be a closed oriented Riemann surface of genus at least 2. By using symplectic chain complex, we construct a volume element for a Hitchin component of Hom(π₁(Σ),PSLₙ(ℝ))/PSLₙ(ℝ) for n > 2.

On almost cosymplectic (−1, μ, 0)-spaces

Piotr Dacko, Zbigniew Olszak (2005)

Open Mathematics

In our previous paper, almost cosymplectic (κ, μ, ν)-spaces were defined as the almost cosymplectic manifolds whose structure tensor fields satisfy a certain special curvature condition. Amongst other results, it was proved there that any almost cosymplectic (κ, μ, ν)-space can be 𝒟 -homothetically deformed to an almost cosymplectic −1, μ′, 0)-space. In the present paper, a complete local description of almost cosymplectic (−1, μ, 0)-speces is established: “models” of such spaces are constructed,...

On almost cosymplectic (κ,μ,ν)-spaces

Piotr Dacko, Zbigniew Olszak (2005)

Banach Center Publications

An almost cosymplectic (κ,μ,ν)-space is by definition an almost cosymplectic manifold whose structure tensor fields φ, ξ, η, g satisfy a certain special curvature condition (see formula (eq1b)). This condition is invariant with respect to the so-called -homothetic transformations of almost cosymplectic structures. For such manifolds, the tensor fields φ, h ( = ( 1 / 2 ) ℒ ξ φ ), A ( = -∇ξ) fulfill a certain system of differential equations. It is proved that the leaves of the canonical foliation of an almost cosymplectic...

On category 𝒪 for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

We study equivalences for category 𝒪 p of the rational Cherednik algebras 𝐇 p of type G ℓ ( n ) = ( μ ℓ ) n ⋊ 𝔖 n : a highest weight equivalence between 𝒪 p and 𝒪 σ ( p ) for σ ∈ 𝔖 ℓ and an action of 𝔖 ℓ on an explicit non-empty Zariski open set of parameters p ; a derived equivalence between 𝒪 p and 𝒪 p ' whenever p and p ' have integral difference; a highest weight equivalence between 𝒪 p and a parabolic category 𝒪 for the general linear group, under a non-rationality assumption on the parameter p . As a consequence, we confirm special cases of conjectures...

On characterization of Poisson and Jacobi structures

Janusz Grabowski, Paweŀ Urbański (2003)

Open Mathematics

We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.

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