Degenerate Poisson structures in dimension 3

Formal normal forms of degenerate Poisson structures in dimension 3 are described. The main tool of the study is a spectral sequence previously introduced by the author. In particular, this method allows one to obtain a new proof of the linearizability of Poisson structures with semisimple linear part. However, there are nonlinearizable Poisson structures in dimension 3 as well.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 579–592, April, 1998.The author wishes to thank the referee for pointing out reference [3] and for other useful remarks.     

Poisson approximations for 2-dimensional patterns

LetX=(X _ij) _n×n be a random matrix whose elements are independent Bernoulli random variables, taking the values 0 and 1 with probabilityq _ij andp _ij (p _ij+q _ij=1) respectively. Upper and lower bounds for the probabilities ofm non-overlapping occurrences of a square submatrix with all its elements being equal to 1, are obtained. Some Poisson convergence theorems are established forn . Numerical results indicate that the proposed bounds perform very well, even for moderate and small values ofn.This work is supported in part by the Natural Science and Engineering Research Council of Canada under Grant NSERC A-9216.     

Singularities of 3-dimensional varieties admitting an ample effective divisor of Kodaira dimension zero

For a normal threefoldX with an effective Cartier divisorH, which is a minimal model of Kodaira dimension zero, we prove that eitherX is a generalized cone overH, orX has quadruple singularities andH is either a K3 surface, or an Enriques surface. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 617–625, April, 1996. The author is extremely grateful to V. A. Iskovskikh and Yu. G. Prokhorov for valuable advice and fruitful discussions. This research was partially supported by the International Science Foundation under grant No. M90 000.     

Quivers and Poisson structures

We produce natural quadratic Poisson structures on moduli spaces of representations of quivers. In particular, we study a natural Poisson structure for the generalised Kronecker quiver with 3 arrows.     

Affine Poisson structures

We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all equivalence classes of central extensions of the Lie algebra by ℝ. We characterize all the affine Poisson structures defined on the duals of some lower dimensional Lie algebras. It is shown that under a certain condition every Poisson structure locally looks like an affine Poisson structure. As an application, we show the role played by affine Poisson structures in mechanics. Finally, we prove some involution theorems.     

Geometry of Poisson structures

The purpose of this paper is to consider certain mechanisms of the emergence of Poisson structures on a manifold. We shall also establish some properties of the bivector field that defines a Poisson structure and investigate geometrical structures on the manifold induced by such fields. Further, we shall touch upon the dualism between bivector fields and differential 2-forms.     

Rigidity of Poisson structures

We study germs of analytic Poisson structures which are suitable perturbations of a quasihomogeneous Poisson structure in a neighborhood of the origin of ℝ ⁿ or ℂ ⁿ , a fixed point of the Poisson structures. We define a “diophantine condition” relative to the quasihomogeneous initial part ie256-1 which ensures that such a good perturbation of 256-2 which is formally conjugate to 256-3 is also analytically conjugate to it.     

Singularities of Jacobi series on C2 and the Poisson process equation

A classical theorem of Gabor Szego relates the singularities of real zonal harmonic expansions with those of associated analytic functions of a single complex variable. Zeev Nehari developed the counterpart for Legendre series on the C-plane by generalizing Szego's theorem. This paper function theretically identifies the singularities of analytic symmetric Jacobi series on C² with those of analytic functions on the C-plane. One feature is that information about the singularities of solutions of Solomon Bochner's Poisson process equation flow from the expansion coefficients. Others are that the Szego and Nehari theorems appear on characteristic subspaces. And, that this PDE, unlike those normally encountered in function theory, is hyperbolic in the real domain.     

Smoothness and Poisson structures of Bridgeland moduli spaces on Poisson surfaces

Let X be a projective smooth holomorphic Poisson surface, in other words, whose anti-canonical bundle is effective. We show that moduli spaces of certain Bridgeland stable objects on X are smooth. Moreover, we construct Poisson structures on these moduli spaces.

     

Normal forms of Poisson structures

In the paper, formal normal forms of Poisson structures are found. As a consequence, obstructions to the formal linearization of Poisson structures are obtained, which gives a generalization of the Weinstein linearization theorem. Lie-Sklyanin algebras corresponding to Poisson structures with trivial linearization are introduced and studied as well. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 220–235, February, 1997. Translated by A. I. Shtern     

Poisson structures on moduli spaces of sheaves over Poisson surfaces

Summary We introduce and study the notion of Poisson surface. We prove that the choice of a Poisson structure on a surfaceS canonically determines a Poisson structure on the moduli space of stable sheaves onS. This result generalizes previous results obtained by Mukai [14], for abelian orK3 surfaces, and by Tyurin [16].Oblatum 13-VI-1994 & 22-III-1995This article was processed by the author using thepjourlm style file from Springer-Verlag     

On characterization of Poisson and Jacobi structures

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.     

Poisson structures on basic cycles

The Poisson structures on a basic cycle are determined completely via quiver techniques. As a consequence, all Poisson structures on basic cycles are inner.     

Poisson fiber bundles and coupling Dirac structures

Poisson fiber bundles are studied. We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also provide some examples.      

动力系统的Poisson结构和可积变形

利用动力系统的守恒积分构造Poisson结构,将动力系统表示为广义Hamilton系统的形式,并以一个三维动力系统为例,通过添加任意可微函数推广守恒积分,构造系统的可积变形,并给出变形后系统的Poisson结构,由此得到了新的刘维尔可积系统.     

Poisson structures transverse to coadjoint orbits

We show that the Poisson structure transverse to a coadjoint orbit in the dual of a semisimple Lie algebra has a polynomial structure matrix, as conjectured by Damianou.