Andriy Panasyuk

ON INTEGRABILITY OF GENERALIZED VERONESE CURVES OF DISTRIBUTIONS

Abstract Given a 1-parameter family of 1-forms γ ( t ) = γ 0 + tγ 1 + ···+ t n ψ n , consider the condition dγ ( t )∧ γ ( t ) = 0 (of integrability for the annihilated by γ ( t ) distribution w ( t )). We prove that in order that this condition is satisfied for any t it is sufficient that it is satisfied for N = n + 3 different values of t (the corresponding implication for N = 2 n + 1 is obvious). In fact we give a stronger result dealing with distributions of higher codimension. This result is related to the so-called Veronese webs and can be applied in the theory of bihamiltonian structures.

Bi-Hamiltonian structures with symmetries, Lie pencils and integrable systems

There are two classical ways of constructing integrable systems by means of bi-Hamiltonian structures. The first one supposes nondegeneracy of one of the Poisson structures generating the pencil and uses the so-called recursion operator. This situation corresponds to the absence of Kronecker blocks in the so-called Jordan–Kronecker decomposition. The second one, which corresponds to the absence of Jordan blocks in this decomposition, uses the Casimir functions of different members of the pencil. In this paper, we consider the general case of a bi-Hamiltonian structure with both Kronecker and Jordan blocks and give a criterion for the completeness of the corresponding family of functions. This result is related to a natural action of some Lie algebra which gives a symmetry of the whole pencil. The criterion is applied to bi-Hamiltonian structures related to Lie pencils, although we also discuss other possible applications.

Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems

Abstract We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group G on a direct product of N coadjoint orbits O =O 1 ×⋯×O N ⊂ g ∗ ×⋯× g ∗ endowed with a bihamiltonian structure whose first generator is the standard symplectic form on O . As a result we get the so-called classical Gaudin system on O . The method works for a wide class of Lie algebras including the semisimple ones and for a large class of orbits including the generic ones and the semisimple ones.

Isomorphisms of Some Complex Poisson Brackets

Given a complex manifold Mi, structures of Poisson algebras on ε(Mi)=C∞(Mi,C) which are associated with a nondegenerate ∂-closed (2,0)-form ωi on Mi are considered. It is shown that every isomorphism of Poisson structures ε(M1) → ε(M2) is generated by a biholomorphic map ψ:M2 → M1 such that ω2 = ψ*ω1

Veronese webs and nonlinear PDEs

Abstract Veronese webs are closely related to bi-Hamiltonian systems, as was shown by Gelfand and Zakharevich. Recently a correspondence between Veronese three-dimensional webs and three-dimensional Einstein–Weyl structures of hyper-CR type was established. The latter were parametrized by Dunajski and Krynski via the solutions of the dispersionless Hirota equation. In this paper we show relations of Veronese three-dimensional webs to several other integrable equations, deform these equations preserving integrability via a dispersionless Lax pair and compute the corresponding contact symmetries, Backlund transformations and Einstein–Weyl structures. Realization of Veronese webs through solutions of these deformed integrable PDE is based on a correspondence between partially integrable Nijenhuis operators to the operator fields with vanishing Nijenhuis tensor. This correspondence could be used to construct a link between bi-Hamiltonian finite-dimensional integrable systems and dispersionless integrable PDE related to the Veronese webs.

The Darboux-Type Theorems for ∂-Symplectic and ∂-Contact Structures

A ∂-symplectic structure on a complex manifold M of complex dimension2n is given by a smooth ∂-closed (2, 0)-form ω such thatωn is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally ω can be represented as∑i=1n ∂ fi∧ ∂ fn+i for appropriate smooth complex valuedfunctions f1, ..., f2n. We also present a contact version of this theorem.