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Integrable systems and group actions

100%
Miranda E.
Open Mathematics
|
2014
|
tom 12
|
nr 2
240-270
EN
The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.
2
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A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

100%
Poon Y. S., Simanyi J.
Complex Manifolds
|
2017
|
tom 4
|
nr 1
137-154
EN
A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.
3
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On localization in holomorphic equivariant cohomology

81%
Bruzzo U., Rubtsov V.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1442-1454
EN
We study a holomorphic equivariant cohomology built out of the Atiyah algebroid of an equivariant holomorphic vector bundle and prove a related localization formula. This encompasses various residue formulas in complex geometry, in particular we shall show that it contains as special cases Carrell-Liebermann’s and Feng-Ma’s residue formulas, and Baum-Bott’s formula for the zeroes of a meromorphic vector field.
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