Symplectic realizations of bihamiltonian

structures

Andriy Panasyuk∗

Division of Mathematical Methods in Physics,University of Warsaw,

Hoza St. 74, 00-682 Warsaw, Poland, e-mail: panas@fuw.edu.pland

Mathematical Institute of Polish Academy of Sciences

Sniadeckich St. 8, 00-950 Warsaw, Poland

To my wife Larysa

0 Introduction

A smooth manifold M is endowed by a Poisson pair if two linearlyindependent bivectors c1, c2 are defined on M and moreover cλ = λ1c1+λ2c2 is a Poisson bivector for any λ = (λ1, λ2) ∈ R2. A bihamiltonianstructure J = {cλ} is the whole 2-dimensional family of bivectors. Thestructure J (the pair (c1, c2)) is degenerate if rank cλ < dim M, λ ∈ R2.

The degenerate bihamiltonian structures play important role in thetheory of completely integrable systems due to the following fact. LetZcλ

denote the set of Casimir functions for cλ and let J0 ⊂ J be thesubfamily formed by the Poisson bivectors of maximal rank. It is easyto see that the functions from F0 = SpanR(

⋃cλ∈J0

Zcλ) are in involution

with respect to any cλ. If F0 contains enough independent functionsthere exists a chance of getting the completely integrable system ([2])on symplectic leaves of cλ. This is possible, for example, in the followingsituation. Consider a fixed Poisson bivector cλ ∈ J0 and a point x onits symplectic leaf S(x) of maximal dimension. Assume that for anycλ differentials of functions from Zcλ

generate the kernel of cλ(x). Thefamily F0 is complete at x with respect to cλ, i.e. defines a lagrangianfoliation in some neighbourhood of x in S(x), if and only if

∗Partially supported by the Polish grant KBN 2 PO3A 135 16.

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(∗) rankC(λ1c1 + λ2c2)(x) is maximal for any λ = (λ1, λ2) ∈ C2 \ {0}.This criterion of A.Brailov and its generalizations can be found in [4].Degenerate bihamiltonian structures J = {cλ} (Poisson pairs (c1, c2))satisfying (∗) on an open dense subset of M will be called complete.

An intensive study of such objects was done by I.M.Gelfand andI.S.Zakharevich ([10], [11], [12]) in a particular case of bihamiltonianstructures in general position on an odd-dimensional M (the correspond-ing Poisson pairs are necessarily degenerate: rank cλ = 2n, λ ∈ R2 \{0},if dimM = 2n+1; they are also complete). In [11] there was introduceda notion of a Veronese web, i.e. a 1-parameter family of 1-codimensionalfoliations such that the corresponding family of annihilators is repre-sented by the Veronese curve in the cotangent space at each point. Itturns out that Veronese webs form a complete system of local invariantsfor bihamiltonian structures of general position. More precisely, it wasproved in [11] that any such structure J = {cλ} in R2n+1 admits a localreduction to a Veronese web VJ and that for any Veronese web V onecan locally construct a bihamiltonian structure J(V) of general positionin R2n+1 with the reduction equal to V. In the real analytic case J andJ(VJ) are isomorphic.

Complete Poisson pairs of higher corank are also of great interest.For example, the argument translation method of A.S.Mishchenko andA.T.Fomenko ([9]) uses the standard linear Poisson bivector c on a dualspace g∗ to a semisimple Lie algebra g and the constant Poisson bivectorc(a), where a ∈ g∗ is a fixed point on some symplectic leaf of maximaldimension for c. The pair (c, c(a)) is a complete Poisson pair with corankequal to rank g.

In this paper a method of constructing the complete bihamiltonianstructures is presented. Essentially, it is the reduction of complex sym-plectic manifolds (M, ω) by a symplectic action of a real Lie group. Thebivectors c′1, c′2 generating the so built bihamiltonian structure J ′ areobtained as the reductions of the real and imaginary parts of the corre-sponding holomorphic Poisson bivector c = ω−1 (it is easy to see thatsuch c′1, c′2 form a Poisson pair, provided that they are linearly indepen-dent). The necessary and sufficient conditions for the completeness ofJ ′ are obtained. The method is illustrated by a series of examples: itis proved that, given a complex semisimple Lie group G and its genericcoadjoint orbit (M, ω) with the standard symplectic form ω, the re-duction of ω with respect to a compact real form G0 ⊂ G produces acomplete bihamiltonian structure (M/G0 necessarily has singularities;

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the formulated result concernes only the smooth part).The paper is organized as follows. In Section 1 we recall some def-

initions and facts from the theory of Poisson manifolds and introducethe class of complex Poisson structures, which are generalizations of thestandard ones to the case of the complexified tangent bundle. Holomor-phic Poisson structures are strictly contained in this class.

Section 2 is devoted to bihamiltonian structures and their relationswith the completely integrable systems. We define complete bihamil-tonian structures and show that they include the mentioned case ofgeneral position.

In Subsections 3.1-3.9 we prove that the Poisson reduction (c′1, c′2)of a Poisson pair (c1, c2) is again a Poisson pair under the require-ment of the linear independence for c′1, c′2. This result follows fromthe natural behavior of the Schouten bracket with respect to the re-duction and is preparational. The main theorem of Section 3 (see4.4) deals with the reductions (M, ω) −→ (M ′, (c′1, c′2)), where (M,ω)are complex symplectic manifolds and c′1, c′2 are the push-forwards ofc1 = Re(ω)−1, c2 = Im(ω)−1 (such (M,ω) are called realizations for(M ′, (c′1, c′2))). It presents the necessary and sufficient conditions forthe completeness of the Poisson pair (c′1, c′2) under the assumption thatthe leaves of the submersion M −→ M ′ are the generic CR-submanifoldsin M . Considering of nongeneric case is also meaningful but will not betouched in this paper. In the end of this section a notion of minimalrealization is discussed.

The goal of Section 4 is to build examples of reductions (M,ω) −→(M ′, (c′1, c′2)) such that (c′1, c′2) are the complete Poisson pairs. The cor-responding complex symplectic manifolds (M,ω) will be coadjoint or-bits of a complex semisimple Lie group G with the standard symplecticstructure. The central result of this section (Theorem 6.7) establishesthe completeness of (c′1, c′2) on the smooth part of the reduction M/G0,where M is a coadjoint orbit of general position and G0 is a compactreal form of G. The proof of this result (Subsection 6.8) requires somepreliminar work that is done in 6.2-6.5. Some of presented there re-sults are devoted to the CR-geometry of the coadjoint G0-orbits andseem to be of independent interest. Subsections ??-?? are intended toexplain the proof as a generalization of the above mentioned methodof argument translation. Moreover the last theorem of Section 4 showsthat this proof works at least locally for arbitrary finite-dimensional Liealgebras with codim Sing g∗ ≥ 2, where Sing g∗ denotes the sum of thecoadjoint orbits of nonmaximal dimension.

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Off course, the inspiration for this paper is the theory of symplecticrealizations for Poisson structures ([19]). The last section contains adiscussion of open questions mainly motivated by references [11] and[19].

that there are no (anti)holomorphic functions constant along K, i.e.that among constants on K there are no Casimir functions of c and c(the only, up to rescaling, degenerate bivectors in the family J). On theinfinithesimal level this

1 Complex Poisson structures and other pre-liminaries

1.1. Let M be a C∞-manifold; denote by E(M) (EC(M)) a space ofC∞-smooth real (complex) valued functions on M . For a C∞ vectorbundle π : N → M , let Γ(N) denote the space of C∞-smooth sectionsof π. Elements of Γ(

∧2 TM) (Γ(∧2 T CM)) will be called (complex)

bivectors for short.

1.2. Definition A (complex) bivector c ∈ Γ(∧2 TM) (Γ(

∧2 T CM)) iscalled Poisson if [c, c] = 0.

Here [ , ] denotes the complex extension of the Schouten bracketwhich associates a trivector field [c1, c2] ∈ Γ(

∧3 T CM) to two bivec-tors c1, c2 ∈ Γ(

∧2 T CM). The corresponding local coordinate formulalooks as follows:

[c1, c2]ijk(x) =12

∑

c.p.ijk(cir

1 (x)∂

∂xrcjk2 (x) + cir

2 (x)∂

∂xrcjk1 (x)), (1.2.1)

where cα = cijα (x) ∂

∂xi ∧ ∂∂xj , α = 1, 2,

∑c.p.ijk denotes the sum over

the cyclic permutations of i, j, k and the summation convention overrepeated indices is used (the latter will be used systematically in thispaper).

1.3. Definition Let M be a complex manifold. A holomorphic sectionof the bundle

∧2 T 1,0M ⊂ T CM will be called holomorphic bivector. Inparticular, holomorphic bivectors can be considered as complex ones andthey will be called holomorphic Poisson if, in addition, they are Poissonin the sense of previous definition.

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1.4. Definition A hamiltonian vector field c(f) corresponding to afunction f ∈ E(M) (EC(M)) is obtained by the contraction of the dif-ferential df and the Poisson bivector c with respect to the first index.

1.5. Proposition A (complex) bivector c is Poisson if and only if anoperation { , }c : E(M)×E(M) −→ E(M) (EC(M)×EC(M) −→ EC(M))given by

{f, g}c = c(f)g, f, g ∈ E(M) (EC(M))

is a Lie algebra bracket over R (C).If c is Poisson, then the map f 7→ c(f) : E(M) → Vect, where

Vect(M) is a Lie algebra of smooth vector fields on M with the commu-tator bracket, is a Lie algebra homomorphism.

1.6. Definition { , }c is called a Poisson bracket corresponding to the(complex) Poisson bivector c. A family of functions F ⊂ E(M) (EC(M))is involutive with respect to c if {f, g}c = 0 for each two functionsf, g ∈ F .

1.7. Definition Let c be a (complex) bivector; set Pc,x = {c(f)(x); f ∈E(M)} ⊂ TxM (Pc,x = {c(f)(x); f ∈ EC(M)} ⊂ T Cx M). A generalizeddistribution of subspaces Pc ⊂ TM (T CM) is said to be a characteristicdistribtution for the bivector c.

1.8. Definition Consider a (complex) bivector c at x ∈ M as a mapc]x : T ∗xM −→ TxM ((TCx M)∗ −→ T Cx M) evaluating the first argument

of the bivector on a 1-form. Kernel ker c(x) and rank rank c(x) of cat x are defined as that of c]

x. We say that c is nondegenerate if c]

is an isomorphism. A complex bivector is called nondegenerate in theholomorphic sense if the restricted sharp map c] : (T 1,0M)∗ −→ T 1,0Mis an isomorphism.

Obviously, Pc,x = c]x(T ∗xM) (Pc,x = c]

x((TCx M)∗)). Note that a com-plex bivector nondegenerate in holomorphic sense is not nondegeneratesince Pc,x ⊂ T 1,0M 6= T CM . We shall usually understand the nonde-generacy of holomorphic bivectors in the holomorphic sense.

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1.9. Theorem ([14]) Let c be a real Poisson bivector. The generalizeddistribution Pc is completely integrable, i.e. there exists a tangent to Pc

generalized foliation {Sα}α∈I on M : TxSα = Pc,x for any α ∈ I andfor any x ∈ Sα. The restriction of c to each Sα is a nondegeneratePoisson bivector; thus Sα are symplectic manifolds with syplectic formsωα = (c|Sα)−1.

Here and subsequently the 2-form ω inverse to a nondegeneratebivector c is defined as follows. If ∧2c] is the extension of the abovedefined sharp map to the second exterior power of T ∗M , then ω =(∧2c])−1(c). The inverse to a nondegenerate 2-form bivector is deinedsimilarly.

The above Theorem is also true in the complex analytic category ifwe understand Pc as a holomorphic subbundle in T 1,0M and the non-degeneracy in the holomorphic sense. The definition of inverse objectsin this case is analogous to real one.

1.10. Definition The submanifolds Sα are called symplectic leaves ofa Poisson bivector c.

1.11. Proposition Given a complex Poisson bivector c ∈ Γ(∧2 T CM),

its characteristic distribution Pc ⊂ TCM is involutive, i.e.

[v, w](x) ∈ Pc,x

for any complex valued vector fields v, w such that v(x), w(x) ∈ Pc,x, x ∈M .

In general, one can say nothing about the complete integrability ofPc even if one understands this in spirit of the Newlander-Nierenbergtheorem. A nonconstant rank of the subspaces Pc,x or Pc,x

⋂P c,x (the

overline means the complex conjugation) may be the obstruction hereas well as some other reasons (see [18]).

1.12. Convention In the sequel, all Poisson bivectors will be assumedto have maximal rank on an open dense subset in M . For real Poissonbivectors this is equivalent to the following: the union of symplecticleaves of maximal dimension is dense in M .

1.13. Definition Let c be a (complex) Poisson bivector. Define rank cas maxx∈M rank c(x).

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1.14. Definition A Casimir function f ∈ E(U) (EC(U)) over an openset U ⊂ M for a (complex) Poisson bivector c is defined by the conditionc(f) = 0. A space of all Casimir functions over U for c is denoted byZc(U) and Zc,x stands for the space of germs of Casimir functions atx, x ∈ M .

Note that if c is real and rank c < dimM there exist local nontrivialCasimir functions and their differentials at x span ker c(x), providedthat x is taken from a symplectic leaf of maximal dimension. This isnot true concerning the global Casimir functions: it is easy to constructa Poisson bivector c with rank c < dimM posessing only trivial ones.

1.15. Definition Let (M,ω), dimM = 2n, be a symplectic manifold.A submanifold L ⊂ M is called

• coisotropic if (TxL)⊥ω(x) ⊂ TxL for any x ∈ L;

• isotropic if (TxL)⊥ω(x) ⊃ TxL for any x ∈ L;

• lagrangian if (TxL)⊥ω(x) = TxL for any x ∈ L.

A foliation L on M is coisotropic (isotropic, lagrangian) if so is itsevery leaf.

Here ⊥ω(x) stands for a skew-orthogonal complement in TxM withrespect to ω(x). For the third case the following definition is equivalent:dimL = n and ω|TL ≡ 0.

1.16. We shall need a specific generalization of this definition in thecomplex case. Let M be a complex manifold with the complex structureJ : TM −→ TM . Consider a C∞-smooth submanifold L ⊂ M . WriteTCR

x L for TxL⋂J TxL and T 1,0

x L for T Cx L⋂

T 1,0x M, x ∈ L. Another

definition for T 1,0x L is the following: T 1,0

x L = {v − iJ v; v ∈ TCRx L}.

1.17. Definition ([18],[3]) L is called a CR-submanifold in M ifdimT 1,0

x L is constant along L; we say that this number is CR-dimensionof L; L is generic (completely real) if TxL + J TxL = TxM (TxL ⊕J TxL = TxM) for each x ∈ L.

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If a generic CR-submanifold L is given by the equations {f1 =α1, . . . , fk = αk}, fi ∈ E(M), such that df1∧. . .∧dfk 6= 0 along L, αi ∈ R,then ∂f1 ∧ . . .∧ ∂fk 6= 0 along L and T 1,0

x L = {∂f1(x), . . . , ∂fk(x)}⊥1,0,where ∂ is the (1,0)-differential on M , ⊥ 1, 0 denotes the annihilator inT 1,0

x M .

1.18. Definition A foliation L on M is a generic (completely real)CR-foliation if its each leaf is a generic (completely real) CR-submanifold.

1.19. Definition Let (M,ω) be a holomorphic symplectic manifold.A CR-submanifold L ⊂ M is

• CR-coisotropic if (T 1,0x L)⊥ω(x) ⊂ T 1,0

x L for any x ∈ L;

• CR-isotropic if (T 1,0x L)⊥ω(x) ⊃ T 1,0

x L for any x ∈ L;

• CR-lagrangian if (T 1,0x L)⊥ω(x) = T 1,0

x L for any x ∈ L.

A CR-foliation L on M is said to be CR-coisotropic (CR-isotropic,CR-lagrangian) if so is its every leaf.

Here ⊥ω(x) denotes a skew-orthogonal complement in T 1,0x M with

respect to the (2,0)-form ω(x).Suppose L is generic and consits of the common level sets of the

functions f1, . . . , fk ∈ E(M), k ≤ n = (1/2) dimCM . Then L is CR-coisotropic if and only if the family {f1, . . . , fk} is involutive with respectto the holomorphic Poisson bivector c = (ω)−1. In particular, if k = none gets CR-lagrangian foliation.

1.20. Definition Let (M, ω), dimM = 2n, be a symplectic manifold.A completely integrable system on M is defined as a family of func-tions F ⊂ E(M) involutive with respect to c = (ω)−1 and containinga subfamily of n functions that are functionally independent almost ev-erywhere on M . In other words, a completely integrable system on Mis a lagrangian foliation L on an open dense subset in M .

We conclude this section by recalling main definitions concerninghamiltonian actions of Lie groups (see [8] for details).

1.21. Definition Let G be a connected Lie group with the Lie algebrag. Assume it is acting on a Poisson manifold M with the Poisson bivec-tor c, i.e. a Lie algebra homomorphism ρ : g ← Vect(M) is given. The

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action is called Hamiltonian if there exists a Lie algebra homomorphismψ : g → E(M) such that the following diagram is commutative

gψ−→ E(M)

‖ ↓ c(·)g

ρ−→ Vect(M),

where c(·) is a Lie algebra homomorphism of taking the hamiltonianvector field (see Proposition 1.5).

The map x 7→ ϕx : M → g∗ defined by ϕx(v) = ψ(v)(x), v ∈ g iscalled the moment map.

2 Bihamiltonian structures and completeness

Let M be a C∞-manifold.

2.1. Definition Two linearly independent (complex) Poisson bivec-tors c1, c2 on M form a Poisson pair if cλ = λ1c1 + λ2c2 is a (complex)Poisson bivector for any λ = (λ1, λ2) ∈ R2 (C2).

2.2. Proposition A pair of linearly independent (complex) Poissonbivectors (c1, c2) is Poisson if and only if [c1, c2] = 0.

2.3. Definition Let M be a C∞-manifold. A (complex) bihamiltonianstructure on M is defined as a two-dimensional linear subspace J ={cλ}λ∈S of (complex) Poisson bivectors on M parametrized by a two-dimensional vector space S over R (C).

It is clear that every Poisson pair generates a bihamiltonian structureand the transition from the latter one to a Poisson pair corresponds toa choice of basis in S. We shall write (J, c1, c2) for a bihamiltonianstructure J with a chosen Poisson pair (c1, c2) generating J .

2.4. Definition Let (J, c1, c2) be a bihamiltonian structure. A com-plex bihamiltonian structure

JC = {λ1c1 + λ2c2; (λ1, λ2) ∈ C2}

is called the complexification of J .

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2.5. Proposition A complex bihamiltonian structure J is the com-plexification of some real one if and only if one can choose a generatingJ Poisson pair (c, c), where c ∈ ∧2(T CM), the bar stands for the com-plex conjugation.

Proof. Generate J ′ be Re c, Im c. Then J = (J ′)C. Conversely, onechecks that for (J ′, c1, c2) the complexification (J ′)C is generated byc1 ± ic2. q.e.d.

2.6. Definition Let J be a (complex) bihamiltonian structure and letJ0 ⊂ J be a subfamily of (complex) Poisson bivectors of maximal rankR0 (the set J \ J0 is at most a finite sum of 1-dimensional subspaces).We say that J is symplectic if rank cλ = dimM for any cλ ∈ J0 andthat J is degenerate otherwise.

2.7. Example Consider a family JC generated by a pair (c, c), wherec = (ω)−1 is a complex Poisson bivector inverse to a holomorphic sym-plectic form ω on a complex symplectic manifold M . Since c is holo-morphic and c is antiholomorphic, we have [c, c] = 0. Thus JC is abihamiltonian structure. By Proposition 2.5 it is the complexificationof the real bihamiltonian structure (J,Re c, Im c). This example is fun-damental for the paper and we shall need the following fact.

2.8. Theorem Let M, ω, c and JC be as in Example 2.7. Then JC issymplectic and the only degenerate bivectors in JC are those proportionalto c and c. Moreover, rankC c = rankC c = 1

2 dimCM, Pc = T 1,0M,Pc =T 0,1M .

Proof. The last assertion is obvious as well as the following equality

c ω = 0,

where stands for the contraction with respect to the first index. Forω1 = Re ω, ω2 = Imω, c1 = Re c, c2 = Im c this implies

c1 ω1 + c2 ω2 = 0, c2 ω1 − c1 ω2 = 0.

We have for λ ∈ C(c1 + λc2) (ω1 − λω2) = c1 ω1 − λ2c2 ω2 − λ(c1 ω2 − c2 ω1) =

(1 + λ2)c1 ω1 = (1 + λ2)14idTM .

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The last equality is verified directly in the Darboux local coordinates.Thus (c1 + λc2)−1 = 4

1+λ2 (ω1 − λω2). Since c1 ω1 = −c2 ω2, c2 is alsonondegenerate. q.e.d.

2.9. Definition Let (M, ω) be a complex symplectic manifold. Thebihamiltonian structure J and its complexification JC from Example 2.7are called holomorphic symplectic.

2.10. Given a (complex) bihamiltonian structure J , let F0 denote thespace SpanR(

⋃c∈J0

Zc(M)). We take Span in order to obtain a vectorspace: a sum of two Casimir functions for different c1, c2 ∈ J0 need notbe a Casimir function. However, SpanR is enough for both the real andcomplex cases.

The following theorem shows how the degenerate bihamiltonian struc-tures can be applied for constructing the completely integrable systems.

2.11. Theorem Let J be a degenerate (complex) bihamiltonian struc-ture on M . A family F0 is involutive with respect to any cλ ∈ J .

Proof. Let c1, c2 ∈ J0 be linearly independent, fi ∈ Zci , i = 1, 2. Then

{f1, f2}cλ= (λ1c1(f1) + λ2c2(f1))f2 = −λ2c2(f2)f1 = 0. (2.11.1)

Now it remains to prove that for any c ∈ J0, fi ∈ Zc, i = 1, 2, one has{f1, f2}cλ

= 0. For that purpose we first rewrite (2.11.1) as

cλ(x)(φ1, φ2) = 0, (2.11.2)

where φi ∈ ker ci(x), i = 1, 2, x ∈ M , and the lefthandside denotes acontraction of the bivector with two covectors. Second, we fix x suchthat rank c(x) = R0 and approximate df2|x by a sequence of elements{φi}∞i=1, φi ∈ ker ci(x), where ci ∈ J0, i = 1, 2, . . . , is linearly indepen-dent with c. Finally, by (2.11.2) we get cλ(x)(df1|x, φi) = 0 and by thecontinuity {f1, f2}cλ

(x) = 0. Since the set of such points x is dense inM , the proof is finished. q.e.d.

In fact this theorem is true for the local Casimir functions (for thegerms of Casimir functions).

2.12. Definition The functions from the family F0 (see 2.10) arecalled (global) first integrals of the bihamiltonian structure J . The fam-ily of functions SpanR(

⋃c∈J0

Zc(U)) (SpanR(⋃

c∈J0Zc,x) is denoted by

F0(U) (F0,x) and its elements are called local first integrals over anopen U ⊂ M (germs of first integrals at x ∈ M).

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In order to obtain a completely integrable system from Casimir func-tions one should require additional assumptions on the bihamiltonianstructure J . Off course, the condition of completeness given below con-cerns the local Casimir functions (in fact their germs) and may be in-sufficient for obtaining the completely integrable system. However, it isof use if the local Casimir functions are restrictions of the global ones(see Example 2.18, below).

Given a characteristic distribution Pc ⊂ TM (T CM) of some (com-plex) Poisson bivector and a point x ∈ M , let P ∗

c,x denote a dual spaceto Pc,x. Any functional φ ∈ T ∗xM ((T Cx M)∗) can be regarded as anelement of P ∗

c,x called the restriction of φ to Pc,x.

2.13. Definition ([4]) Let J be a (complex) bihamiltonian structure;fix some cλ ∈ J .

J is called complete at a point x ∈ M with respect to cλ if a linearsubspace of P ∗

cλ,x generated over R (C) by the differentials of the germsf ∈ F0,x restricted to Pcλ,x has dimension 1

2 dimR Pcλ,x (12 dimC Pcλ,x).

2.14. Proposition A (complex) bihamiltonian structure J is com-plete with respect to cλ ∈ J at a point x ∈ M if and only if dim(

⋂c∈J0

Pc,x)= 1

2 dimPcλ,x.

Proof is obvious.The following theorem is due to A.Brailov (see [4], Theorem 1.1 and

Remark after it).

2.15. Theorem A (complex) bihamiltonian structure J is completewith respect to cλ ∈ J0 at a point x ∈ M such that Pcλ,x is of maximaldimension if and only if the following condition holds

(∗) rank c(x) = R0 for any c ∈ JC \ {0} (J \ {0}),where R0 is as in 2.6.

Proof of this theorem is a consequence of the following linear algebraicfact.

2.16. Proposition ([4]) Let V be a vector space over R (C) and letJ be a two dimensional linear subspace in

∧2 V . In the real case we letJC ⊂ ∧2 V C denote the complexification of the subspace J . We writeJ0 ⊂ J for the subset of bivectors of maximal rank R0 and F0 ⊂ V ∗

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for the subspace generated by the kernels of bivectors from J0. Let c] :V ∗ → V stand for the corresponding sharp map of c ∈ ∧2 V (cf. 1.8).

Then, given a bivector cλ ∈ J0, the following two conditions areequivalent:

(i) dim(F0|Pλ) = 1/2 dimPλ, where F0|Pλ

= Span{{f |Pλ}f∈F0} and

Pλ = c]λ(V ∗);

(ii) rank c = R0 for any c ∈ JC \ {0} (J \ {0}).

Proof. We reproduce the proof from [4] with a small completion.Condition (i) is equivalent to the following: the skew-orthogonal

complement F⊥cλ0 = (c]

λ(F0))⊥ (annihilator in the sense of the dual pair(V, V ∗)) coincides with F0 (by Proposition 2.11 F0 ⊂ F⊥cλ

0 ).Indeed, in view of Proposition 2.11 the subspace F0|Pλ

is isotropic in(Pλ)∗ with respect to cλ. Thus (i) holds if and only if the subspace F0|Pλ

is lagrangian in (Pλ)∗ with respect to cλ, i.e. the subspace (F0|Pλ)⊥cλ =

(c]λλ (F0|Pλ

))⊥λ , where c]λλ : P ∗

λ → Pλ and the annihilator ⊥λ is in thesense of the dual pair (Pλ, P ∗

λ ), coincides with F0|Pλ. But it is easy

to see that there exists a subspace Fλ ⊂ V ∗ such that Fλ∼= F0|Pλ

via the canonical projection V ∗ → P ∗λ and F0 = Fλ ⊕ ker cλ. Hence

c]λλ (F0|Pλ

) = c]λ(F0); denoting this subspace by W and setting 2N =

dimPλ, r = dim ker cλ one calculates dimensions. The subspace F0|Pλis

lagrangian if and only if dimW = N if and only if dimW⊥ = N + r =dimF0. So we proved the claim.

We continue the proof in the following three steps.First, we observe that for any two nontrivial bivectors a, b ∈ J one

has the equality a](F0) = b](F0). Indeed, suppose that a, b are linearlyindependent. The subspace F0 is generated by a finite number of kernelsker b1, . . . , ker bs, bi ∈ J0. Without loss of generality, we may assumethat bi = αia+βib, where αi, βi 6= 0. Since (αia

] +βib])(ker bi) = 0, i =

1, . . . , s, then a](ker bi) = b](ker bi) and, consequently, a](F0) = b](F0).In the second step we consider the skew-orthogonal complement

F0 = (F0)⊥b = (b](F0))⊥ and note that: 1) it does not depend onb ∈ J \ {0} (previous step); 2) F0 ⊂ F0 (again Proposition 2.11); 3)if a ∈ J0, then b](F0) ⊂ a](F0) for any b ∈ J (this is equivalent to(F⊥b

0 )⊥b ⊃ (F⊥a0 )⊥a or F0 + ker b ⊃ F0 + ker a = F0).

Finally, given two linearly independent bivectors a, b ∈ J , withrank a = R0, we define a ”recursion” operator Φ : F0/F0 → F0/F0 bythe formula Φ(π(ξ)) = π((a])−1b](ξ)), where ξ ∈ F0 and π : F0 → F0/F0

13

is the natural projection. The operator is correctly defined due to theconditions a(F0) = b(F0), b(F0) ⊂ a(F0), and ker a ⊂ F0. It is easyto see that the eigenvalues of Φ are precisely those λ ∈ C for whichrank(a − λb) < R0. In particular (ii) holds if and only if Φ does nothave eigenvalues, i.e. F0 = F0. q.e.d.

Theorem 2.15 shows that J is complete with respect to a fixed cλ ∈J0 at a point x such that the dimension Pcλ,x is maximal if and onlyif J = J0

⋃{0} and J is complete at x with respect to any nontrivialcλ ∈ J . This motivates the next definition.

2.17. Definition Let (J, c1, c2) be a (complex) bihamiltonian struc-ture. The structure J (the pair (c1, c2)) is complete at a point x ∈ M ifcondition (∗) of Theorem 2.15 holds at x. J ((c1, c2)) is called completeif it is so at any point from some open and dense subset in M .

2.18. Example (Method of argument translation, see [9], [4].) Let g

be a nonabelian Lie algebra, g∗ its dual space. Fix a basis {e1, . . . , en} ing with the structure constants {ck

ij}; write {e1, . . . , en} for a dual basisin g∗. The standard linear Poisson bivector on g∗ is defined as

c1(x) = ckijxk

∂

∂xi∧ ∂

∂xj,

where {xk} are linear coordinates in g∗ corresponding to {e1, . . . , en}.In more invariant terms c1 is described as dual to the Lie-multiplicationmap [ , ] : g ∧ g −→ g. It is well-known that the symplectic leaves ofc1 are the coadjoint orbits in g∗. Now define c2 as a bivector withconstant coefficients c2 = c(a), where a is a fixed point on any leafof maximal dimension. It turns out that c1, c2 form a Poisson pairand it is easy to describe the set I of points x for which condition(∗) fails. Consider the complexification (g∗)C ∼= (gC)∗ and the sumSing(gC)∗ of symplectic leaves of nonmaximal dimension for the complexlinear bivector ck

ijzk∂

∂zi∧ ∂

∂zj, where zj = xj + iyj , j = 1, . . . , n, are the

corresponding complex coordinates in (g∗)C. Then I is equal to theintersection of the sets g∗ ⊂ (g∗)C and a, Sing(gC)∗, where a,Sing(gC)∗denotes a cone of complex 2-dimensional subspaces passing through aand Sing(gC)∗. In particular, (c1, c2) is complete for a semisimple g.Note that this gives rise to completely integrable systems, since thelocal Casimir functions on g∗ are restrictions of the global ones, i.e. theinvariants of the coadjoint action.

14

2.19. Example (Bihamiltonian structure of general position on anodd-dimensional manifold, see [11].) Consider a pair of bivectors (a1, a2),ai ∈

∧2 V, i = 1, 2, where V is a (2m + 1)-dimensional vector space;(a1, a2) is in general position if and only if is represented by the Kro-necker block of dimension 2m + 1, i.e.

a1 = p1 ∧ q1 + p2 ∧ q2 + · · ·+ pm ∧ qm

a2 = p1 ∧ q2 + p2 ∧ q3 + · · ·+ pm ∧ qm+1(2.19.1)

in an appropriate basis p1, . . . pm, q1, . . . , qm+1 of V . A bihamiltonianstructure J on a (2m + 1)-dimensional M is in general position if andonly if the pair (c1(x), c2(x)) is so for any x ∈ M . Such J is complete:it is easy to prove that J = J0

⋃{0}, dim⋂

c∈J Pc(x) = n and then useProposition 2.14. In general, a complete Poisson pair at a point is thedirect sum of the Kronecker blocks and the zero pair as the corollaryof the next theorem shows. This theorem is a reformulation of theclassification result for pairs of 2-forms in a vector space ([10], [12]).

2.20. Theorem Given a finite-dimensional vector space V over C anda pair of bivectros (c1, c2), ci ∈

∧2 V, there exists a direct decompositionV = ⊕k

j=1Vj , ci =∑k

j=1 c(j)i , c

(j)i ∈ ∧2 Vj , i = 1, 2, such that each pair

(c(j)1 , c

(j)2 ) is from the following list:

(a) the Jordan block: dimVj = 2nj and in an appropriate basis of Vj

the matrix of c(j)i is equal to

(0 Ai

−ATi 0

), i = 1, 2,

where A1 = Inj (the unity nj × nj-matrix) and A2 = Jλnj

(theJordan block with the eigenvalue λ);

(b) the Kronecker block: dimVj = 2nj + 1 and in an appropriate basisof Vj the matrix of c

(j)i is equal to(

0 Bi

−BTi 0

), i = 1, 2,

where B1 =

100. . .00010. . .00

. . .000. . .10

, B2 =

010. . .00001. . .00

. . .000. . .01

((nj+1)×nj-matrices).

15

(c) the trivial Kronecker block: dimVj = 1, c(j)1 = c

(j)2 = 0;

2.21. Corollary Let J be a (complex) bihamiltonian structure. It iscomplete at a point x ∈ M if and only if a pair (c1(x), c2(x)), ci(x) ∈∧2(TCx M), i = 1, 2, does not contain the Jordan blocks in its decomposi-tion.

Proof follows from the definition of completeness.The following example of a complete Poisson pair shows that the

structure of decomposition to the Kronecker blocks may change frompoint to point.

2.22. Example ( [17]) Let M = R6 with coordinates (p1, p2, q1, . . . , q4),c1 = ∂

∂p1∧ ∂

∂q1+ ∂

∂p2∧ ∂

∂q2, c2 = ∂

∂p1∧ ( ∂

∂q2+ q1

∂∂q3

) + ∂∂p2

∧ ∂∂q4

. Herewe have: two 3-dimensional Kronecker blocks on M \H,H = {q1 = 0};the 5-dimensional Kronecker block and the 1-dimensional zero block onthe hyperplane H.

2.23. Remark The decomposition of a pair of bivectors c1, c2 in avector space V to Kronecker blocks is defined noncanonically. For exam-ple, let us consider 4-dimensional V = Span{e, p, q1, q2}, c1 = p∧q1, c2 =p ∧ q2. Here V = V1 ⊕ V2, where V1 = Span{e}, V2 = Span{p, q1, q2},but instead V1 one can choose any direct complement to V2. However,dimensions of the direct sums for the Kronecker blocks of equal dimen-sion are invariants (see [13],[17]). For instance, dimension of the sumof the trivial Kronecker blocks is equal to dim ker c1 ∩ ker c2) (see theproof of Proposition 2.24, below).

We conclude the section by a result that will be used later on.

2.24. Proposition Let V be a vector space over C and let a pair ofbivectors c1, c2 ∈

∧2 V be such that there are no Jordan blocks in thedecomposition of Theorem 2.20. Set

µ = dim(ker c1 ∩ ker c2)µλ = dim(ker c1 ∩ ker cλ),

where cλ = λ1c1 + λ2c2, λ1, λ2 6= 0. Then µ = µλ.

16

Proof. Let (V ′, a1, a2), V ′ ⊂ V, dimVj = 2m + 1, ai ∈∧2 V ′, be a

nontrivial Kronecker block. By formula 2.19.1

ker aλ = λm2 qm+1 − λm−1

1 λ2qm + · · ·+ (−1)mλm

2 q1,

where aλ = λ1a1+λ2a2 and q1, . . . , qm+1 is a part of the basis p1, . . . , pm,q1, . . . , qm+1 in V ′∗ dual to p1, . . . pm, q1, . . . , qm+1 (let us denote thesebases by p.,q. and p.,q., correspondingly, and call them adapted tothe pair a1, a2). The above formula shows that ker a1 ∩ ker aλ = {0} ifλ2 6= 0.

Now, let V = ⊕kj=1Vj , ci =

∑kj=1 c

(j)i , i = 1, 2, be the decomposi-

tion to the Kronecker blocks and let Vk′+1, . . . , Vk be all trivial ones.Consider a basis of V of the following form

p.(1),q.(1), . . . ,p.(k′),q.(k

′), r1, . . . , rk−k′ ,

where p.(j),q.(j) is a basis of Vj adapted to c(j)1 , c

(j)2 , j = 1, . . . , k′ and

r1, . . . , rk−k′ generate Vk′+1, . . . , Vk, respectively. The dual basis will beof the form

p.(1),q.(1), . . . ,p.(k′),q.(k′), r1, . . . , rk−k′

and the above considerations show that ker c1 ∩ ker cλ is generated byr1, . . . , rk−k′ if λ1 6= 0. q.e.d.

3 Reductions and realizations of bihamiltonianstructures.

Our first aim is to prove that a Poisson reduction of a bihamiltonianstructure is again a bihamiltonian sturcture. This result follows from anatural behavior of the Schouten bracket with respect to the reduction.

3.1. Consider a C∞-smooth surjective submersion p : M −→ M ′ suchthat p−1(x′) is connected for any x′ ∈ M ′. The foliation of its leaveswill be denoted by K. Write p∗ : TM −→ TM ′ (pC∗ : T CM −→ T CM ′)for the corresponding tangent bundle morphism,

∧k p∗ :∧k TM −→∧k TM ′ for its exterior power extension and ker

∧k p∗ for a subbundlein

∧k TM that is a kernel of∧k p∗ Multivector fields on M or M ′ will

be called multivectors for short.

17

If (U, {x1, . . . , xl, y1, . . . , ym′}) is a local coordinate system on Msuch that m′ = dimM ′ and y1, . . . , ym′

are constant along K, then therestriction Z|U of Z ∈ Γ(

∧k TM) belongs to Γ(ker∧k p∗)(U) if and

only if each term of its decomposition with respect to {∂x1 , . . . , ∂xl , ∂y1 ,. . . , ∂ym′} contains at least one ∂xi , 1 ≤ i ≤ l.

3.2. Theorem Let Z ∈ Γ(∧k TM). The following conditions are

equivalent:

(i) LXZ ∈ Γ(ker∧k p∗) ∀X ∈ Γ(ker p∗), where LX is a Lie derivation;

(ii) φXt,∗Z − Z ∈ Γ(ker

∧k p∗) ∀t∀X ∈ Γ(ker p∗), where φXt denotes the

flow of the vector X;

(iii) in any local coordinate system (U, {x1, . . . , xl, y1, . . . , ym′}) on Msuch that m′ = dim M ′ and y1, . . . , ym′

are constant on the leavesof p the multivector Z can be written as

Z(x, y) = Z ′(y) + Z(x, y), (3.2.1)

whereZ ′(y) = Z ′i1...ik(y)∂yi1 ∧ . . . ∧ ∂yik (3.2.2)

and Z ∈ Γ(ker∧k p∗)(U).

If one of these conditions is satisfied for Z, then Z ′(x′) =∧k p∗(Z(x)),

x′ ∈ M ′, x ∈ p−1(x′), is a correctly defined multivector on M ′. More-over, if (p(U), {y1, . . . , ym′}) is the induced local coordinate system onM ′, then the corresponding local expression for Z ′ coincides with (3.2.2).

Proof. In order to prove the last assertion it is sufficient to note thatfor any two points x1, x2 ∈ p−1(x) there exist X1, . . . , Xs ∈ Γ(ker p∗)and t1, . . . , ts ∈ R such that φX1

t1 ◦ · · · ◦ φXsts (x1) = x2 and then use the

second condition.Obviously, (ii) ⇒ (i). To prove the converse we choose a vector

bundle direct decomposition TM = ker p∗ ⊕ C such that Z ∈ Γ(C) ifZ 6∈ Γ(ker p∗) and C is arbitrary otherwise. Let Π : Γ(TM) −→ Γ(C)be a projection on Γ(C) along Γ(ker p∗). Then

d

dtΠ(φX

t∗Z − Z) = Πd

dt(φX

t∗Z − Z) = Π(−φXt∗[X,Z]) = 0

18

(we have used the equality ddtφ

Xt∗Z = −φX

t∗[X, Z] and the fact that[X,Z] = LXZ, see [16]). Thus Π(φX

t∗Z − Z) is a constant with respectto t multivector and, since Π(φX

t∗Z−Z)|t=0 = Π(0) = 0, we deduce thatΠ(φX

t∗Z − Z) ≡ 0.The equivalence (i) ⇔ (iii) follows from the local expression

[X, Z]i1...ik =1k!

εi1...iks1...sk

Xr∂rZs1...sk − 1

(k − 1)!εi1...ikis2...sk

Zrs2...sk∂rXi

(3.2.3)for the Schouten bracket ([16]). Indeed, if one applies (3.2.3) to thelocal coordinate system from condition (iii) one finds that LXZ ∈Γ(ker

∧k p∗) if and only if (3.2.1) holds. q.e.d.

3.3. Definition We say that a multivector Z ∈ Γ(∧k TM) is pro-

jectable or admits the push-forward if one of the conditions of Theorem3.2 is satisfied. The push-forward, which will be denoted by Z ′, is auniquely defined multivector from Γ(

∧k TM ′), see Theorem 3.2.

3.4. Definition A complex multivector Z ∈ Γ(∧k T CM) admits the

push-forward Z ′ ∈ Γ(∧k T CM ′) if the multivectors ReZ, ImZ ∈

Γ(∧k TM) do so and Z ′ = (ReZ)′ + i(ImZ)′.

3.5. Corollary Let c be a (complex) bivector on M admitting thepush-forward c′ ∈ Γ(

∧2 TM ′) (c′ ∈ Γ(∧2 TCM ′)). Then for any x′ ∈ M ′

and any x ∈ p−1(x′) the following conditions hold:

(i) the subspace p∗,x(]((TxK)⊥) ⊂ Tx′M′ (pC∗,x(]((TCxK)⊥) ⊂ T Cx′M

′),where ⊥ is the annihilator sign, is independent of x;

(ii) the kernel of the map p∗,x|c]((TxK)⊥) (pC∗,x|c]((T CxK)⊥)) equalsc]((TxK)⊥) ∩ TxK (c]((TCxK)⊥) ∩ T CxK);

(iii)

Pc′,x′ ∼= c]((TxK)⊥)/(c]((TxK)⊥) ∩ TxK)(Pc′,x′ ∼= c]((TCxK)⊥)/(c]((TCxK)⊥) ∩ T CxK)).

Proof. iii) follows from i) and ii). These last are consequences ofTheorem 3.2, condition iii). q.e.d.

19

3.6. Remark Although dim c]((TxK)⊥)/c]((TxK)⊥)∩TxK is constantalong K, dimensions of c]((TxK)⊥) and c]((TxK)⊥) ∩ TxK may not beso. For example, let p : R4 → R3 be the projection(x0, x1, x2, x3) 7→(x1, x2, x3) and let c = x0∂x0 ∧ ∂x1 + ∂x2 ∧ ∂x3 . Then c is projectable,but dimension of c]((TxK)⊥) = Span{∂x2 , ∂x3 , x0∂x0} jumps at 0.

3.7. Theorem Let a bivector Zi ∈ Γ(∧2 TM) admit the push-forward

Z ′i ∈ Γ(∧2 TM ′), i = 1, 2. Then a trivector Z = [Z1, Z2] ∈ Γ(

∧3 TM)admits the push-forward Z ′ ∈ Γ(

∧3 TM ′) and Z ′ = [Z ′1, Z ′2].

Proof. In any local coordinate system as in condition (iii) of Theorem3.2 Zi can be written in the form

Zi(x, y) = Z ′i(y) + Zi(x, y),

where Z ′i(y) = Z ′jki (y)∂yj ∧ ∂yk and Zi ∈ Γ(ker∧2 p∗)(U). By formula

(1.2.1)[Z1, Z2](x, y) = [Z ′1, Z

′2](y) + Z(x, y),

where Z ∈ Γ(ker∧3 p∗)(U). Thus by Theorem 3.2 Z admits the push-

forward Z ′ and Z ′ = [Z ′1, Z ′2]. q.e.d.

3.8. Corollary Let (c1, c2) be a Poisson pair on M such that ci ad-mits the push-forward c′i ∈ Γ(

∧2 TM ′), i = 1, 2, and c′1, c′2 are linearlyindependent. Then (c′1, c′2) is a Poisson pair on M ′.

Proof follows immediately from Proposition 2.2 and Theorem 3.2.q.e.d.

3.9. Corollary Let (M,ω) be a complex symplectic manifold and let Gbe a real Lie group acting on M by biholomorphic symplectomorphisms.Assume that M/G is a manifold. Then c1 = Re(ω−1), c2 = Im(ω−1)(see Example 2.7) admit the push-forwards c′1, c′2 ∈ Γ(

∧2 T (M/G)) and(c′1, c′2) is a Poisson pair on M/G, provided that c′1, c′2 are linearly in-dependent.

Proof. It is sufficient to observe that: a) LXjci = 0, i = 1, 2, forgenerators X1, . . . , Xl ∈ ΓTM of the G-action; b) an arbitrary vectorX ∈ Γ(kerp∗), where p : M −→ M/G is a natural projection, is ex-pressed as X = ajXj for some aj ∈ E(M) and LXci = [ajXj , ci] =[aj , ci] ∧Xj ∈ Γ(ker

∧2 p∗), i = 1, 2 (we have used the standard proper-ties of the Schouten bracket, see [16],p.454). q.e.d.

20

3.10. Definition Let p : M −→ M ′ be as in 3.1. A bihamiltonianstructure (J, c1, c2) on M is called projectable (via p) if the bivectorsc1, c2 are so and their push-forwards c′1, c′2 are linearly independent. Thebihamiltonian structure generated by c′1, c′2 on M ′ will be denoted by J ′

and will be called the push-forward or reduction of J .

3.11. Definition Let p : M −→ M ′ be as in 3.1 and let J be a pro-jectable bihamiltonian structure. We say that the triple (M, J,K) is arealization of J ′. If, moreover, J is (holomorphic) symplectic (Deini-tions 2.6, 2.9), we call (M, J,K) (holomrphic) symplectic realization.

4 From symplectic to complete

Let p : M −→ M ′ be as in 3.1 and let J be a projectable symplecticbihamiltonian structure on M with the push-forward J ′. In this sectionwe discuss some conditions on the triple (M, J,K) that guarantee thecompleteness of J ′.

By Corollary 3.5, iii) and by the definition of completeness (2.17)our considerations should start from linear algebra.

4.1. So let V be a vector space over C and let c1, c2 ∈∧2 V be such that

the bihamiltonian structure J = {cλ}λ∈C2 , cλ = λ1c1+λ2c2, λ = (λ1, λ2),where ci is consiedred a constant complex bivector field, is symplectic(Definition 2.6). Also, let K ⊂ V be a subspace such that the push-forwards c′1, c′2 ∈

∧2(V/K) (via the canonical projection p : V → V/K)are linearly independent.

Set R0 = maxλ∈C2 rank cλ, R′0 = maxλ∈C2 rank c′λ and

dλ = dim c]λ(K⊥)/(K ∩ c]

λ(K⊥)).

Write Λ1 = SpanC{λ1}, . . . Λs = SpanC{λs} for the complex lines in C2

on which rank cλ is less than maximal.

4.2. Proposition The condition of completeness

(∗∗) rank c′λ = R′0 for any λ ∈ C2 \ {(0, 0)}

holds if and only if dλ is independent of λ ∈ C2 \ {(0, 0)}.

Proof follows from Corollary 3.5. q.e.d.

21

4.3. Corollary Set Λ =⋃s

i=1 Λi and kλ = dimK ∩ cλ](K⊥). As-sume that K⊥ ∩ ker c]

λi= {0}, i = 1, . . . , s. Then the condition (∗∗) of

Proposition 4.2 holds if and only if

(i) kλ = k is constant over λ ∈ C2 \ Λ;

(ii) kλ1= · · · = kλs

= k.

Proof. Since K⊥∩ker c]

λi= {0}, then dim c]

λi(K⊥) = dim c]

λ(K⊥), i =1, . . . , s, where λ 6∈ Λ. Thus conditions (i), (ii) are equivalent to theconstancy of dλ over λ 6= 0. q.e.d.

The following theorem gives the necessary and sufficient conditionsfor the completeness of the reduction J ′ of a holomorphic symplecticbihamiltonian structure J under an additional assumption. Namely,the foliation K of the leaves of the projection p is supposed to be ageneric CR-foliation.

Let λ1 = 1 + i, λ2 = 1 − i and let Λ denote the cross SpanC{λ1} ∪SpanC{λ2} ⊂ C2. Given a 2-form ω ∈ Γ(

∧2(T CM)∗), write ωλ forλ1 Re ω + λ2 Imω.

4.4. Theorem Let (M,ω) be a complex symplectic manifold with thecorresponding holomorphic symplectic bihamiltonian structure J (seeDefinition 2.9) and let p : M −→ M ′ be as in 3.1. Assume that thefoliation K is a generic CR-foliation on M and that c = ω−1 admitsthe push-forward c′ ∈ Γ(

∧2 T CM ′) linearly independent with c′. Forx′ ∈ M ′, x ∈ p−1(x′), and λ ∈ C2 \ Λ set

kxλ = dim T CxK ∩ (TCxK)⊥ωλ(x)

kx = dimT 1,0x K ∩ (T 1,0

x K)⊥ω(x)

Assume that these numbers are constant along K (cf. Remark 3.6) andset kx′

λ = kxλ, kx′ = kx.

Then the reduction J ′ of J via p is complete at a point x′ ∈ M ′ ifand only if

(i) kx′λ = k is constant in λ;

(ii) kx′ = k

(iii) k = miny′∈M ′ ky′λ .

22

Proof. If W is a real vector space with a complex structure J andY ⊂ W a subspace, let W 1,0 denote the space {w− iJw; w ∈ W} ⊂ W C

and let Y 1,0 = Y C ∩W 1,0 = {y − iJ y; y ∈ Y ∩ J Y } (cf. 1.16).Put V = T Cx M, K = T CxK. We claim that the assumptions of

Corollary 4.3 are satisfied. Indeed, by Theorem 2.8 Λ is appropriatesince the only, up to rescaling, degenerate bivectors from family J arec and c. On the other hand the condition TxK + J TxK = TxM ofCR-genericity for K implies that (TxK)⊥ ∩ J ∗((TxK)⊥) = {0}, whereJ ∗ : T ∗M → T ∗M is adjoint to the complex structure operator J onTM . This means equalities ((TxK)⊥)1,0 = {0} = ((TxK)⊥)0,1 equivalentto K⊥ ∩ T 1,0

x M = 0 = K⊥ ∩ T 0,1x M . Recalling that T 1,0

x M = ker c andT 0,1

x M = ker c we get the claim.Now, put kλ = kx′

λ and kλ1= kλ2

= kx′ and apply Corollary 4.3.Conditions (i) and (ii) are equivalent to the constancy of rank for cλ ∈J ′(x′), λ 6= 0. Its maximality is guaranteed by (iii). q.e.d.

4.5. Corollary In the assumptions of Theorem 4.4 suppose that K iscompletely real (Definition 1.18). Then J ′ is not complete.

Proof. Assume the contrary. By condition (ii) corank of any c′ ∈J ′ \{0} is 0. This contradicts with the definition of completeness. q.e.d.

Given a complete bihamiltonian structure J ′ on M ′, consider itsall realizations with K being a generic CR-foliation. Then the smallestrealizations in this class will be characterized by the biggest intersectionT 1,0K⋂

(T 1,0K)⊥ω.

4.6. Definition Let J ′ be a complete bihamiltonian structure on M ′.Its realization (M, ω) is called minimal if T 1,0K⋂

(T 1,0K)⊥ω = T 1,0K,i.e. K is a CR-isotropic foliation (Definition 1.19).

We shall give another characterization of the minimal realizationsbelow.

4.7. There is a natural CR-coisotropic foliation L ⊃ K associated withany realization J on (M,ω) of a complete J ′. This foliation is builtas follows. Consider the ”real form” J ′R of J ′, i.e. the following realbihamiltonian structure on M ′

{λ1 Re c′ + λ2 Im c′}λ=(λ1,λ2)∈R2 ,

where c′ = p∗c, c = (ω)−1. Now take the foliation L′ =⋂

λ∈R2\{0} S ′λ,S ′λ being the foliation of the symplectic leaves for c′λ ∈ J ′R. Since J ′

23

is complete, L′ is a lagrangian foliation (see Proposition 2.14). Theequations for L′ are the functions from the involutive family F ′0 (see2.10). We define L as p−1(L′). Note that it is CR-coisotropic due tothe fact that its equations f ∈ p−1(F ′0) are in involution with respectto c.

4.8. Theorem A realization J of a complete bihamiltonian structureJ ′ is minimal if and only if the foliation L is a CR-lagrangian foliation.

Proof. Let 2n, r denote rank and corank of the bivector c′ ∈ J ′,respectively, and let dimCM = 2N . By Definiton 4.6 and Theorem?? J is minimal if and only if r = d, where d is CR-dimension of theleaves of K. On the other hand, since K is generic, CR-codimension ofthe leaves is equal to their real codimension, hence 2N − d = 2n + r.Thus the minimality of J is equivalent to the equality n + r = N thatis necessary and sufficient for L to be CR-lagrangian.

5 Canonical complex Poisson pair associatedwith complexification of Lie algebra

5.1. Let g0 be a nonabelian Lie algebra over R and let g = gC0 be itscomplexification. All our further results can be formulated and provedusing g0, g only. But we introduce the corresponding Lie groups for theconvenience.

So let G0 be a connected simply connected Lie group with the Liealgebra Lie(G0) = g0 and let G be a connected simply connected com-plex Lie group with Lie(G) = g. One can consider G0 as a real Liesubgroup in G (see [5], III.6.10).

Let g∗0, g∗ stand for the dual spaces. Fix a basis e1, . . . , en in g0; let ckij

be the corresponding structure constants and let z1 = x1 +iy1, . . . , zn =xn + iyn be the complex linear coordinates in g∗ associated to the dualbasis in g∗ ⊃ g∗0. There are the standard linear bivectors c = ck

ijzk∂

∂zi∧

∂∂zj

in g∗ and c0 = ckijzk

∂∂xi

∧ ∂∂xj

in g∗0. They can be defined intrinsicallyfor instance as the maps

g∗ c−→ g∗∧

g∗, g∗0c−→ g∗0

∧g∗0

dual to the Lie brackets [ , ] : g∧

g −→ g and [ , ]0 : g0∧

g0 −→ g0.

24

It is well-known that the symplectic leaves of c0 (c) are the coadjointorbits for G0 (G). Also, there is a natural action of G0 on g∗:

g(a + ib) = Ad∗g(a) + iAd∗g(b), g ∈ G0, a, b ∈ g0.

Let Sing g∗, Sing g∗0 be unions of symplectic leaves of nonmaximaldimension for c, c0, respectively.

5.2. Proposition The sets Sing g∗, Sing g∗0 are algebraic. Moreover,Sing g∗ is the complexification of Sing g∗0, i.e. the defining polynomi-als for Sing g∗ have real coefficients and are obtained from the defin-ing polynomials of Sing g∗0 by substitution {x1, . . . , xn} 7→ {z1, . . . , zn}.Consequently, codimC Sing g∗ ≤ codimR Sing g∗0.

Proof. These polynomials are minors of m-th order of the n×n-matrix||ck

ijzk||, where m = rank c. q.e.d.

5.3. Convention In the sequel we shall assume that the nonabelianLie algebra g0 satisfies condition codimR Sing g0 ≥ 2.

5.4. This condition is satisfied by a wide class of Lie algebras includ-ing the semisimple ones. Indeed, in the semisimple case we can identifyg∗ and g by means of the Killing form. On the other hand, it is wellkonwn that the algebraic set of all nonregular (regular means semisim-ple contained in the unique Cartan subalgebra) elements is at least ofcodimension three and contains Sing g∗.

5.5. Definition Let us introduce a set

C = {z ∈ g∗; ∃(λ1, λ2) ∈ C2 \ {(0, 0)}, λ1z + λ2z ∈ Sing g∗},

where the bar stands for the complex conjugation corresponding to g0 ⊂g, and call it the incompleteness set (see 5.9 for the explanation of thisterminology).

5.6. Proposition The incompleteness set C is a real algebraic set ofpositive codimension.

25

Proof. We use the product Π = g∗×(C2\{(0, 0)}) with the coordinatesz1, . . . , zn, λ1, λ2 and the real algebraic map φ : Π −→ g∗ given by theformula

(z1, . . . , zn, λ1, λ2) 7→ (λ1z1 + λ2z1, . . . , λ1zn + λ2zn).

The set C can be regarded as pr1(φ−1(Sing g∗)), where pr1 is the pro-jection onto g∗.

To prove that C is of positive codimension we consider C as the unionof all complex 2-dimensional subspaces Sa,b ⊂ g∗ that are generated bypairs a, b ∈ g∗0 and have a nonzero intersection with Sing g∗. If S = Sa,b

is such a subspace, then S = SC0 for S0 = {λ1a+λ2b; (λ1, λ2) ∈ R2} andS

⋂Sing g∗ can be described by the same equations with real coefficients

as S0⋂

SingG0g∗0 in g∗0.

On the other hand, since codim Sing g∗0 ≥ 2, the cone

a,Sing g∗0

of all S0 3 a, S0⋂

Sing g∗0 6= {0}, has codimension at lest 1 in g∗0. Thusfor a fixed a ∈ g∗0 \ Sing g∗0 one can always find b ∈ g∗0 such that thesubspace S0 ⊂ g∗0 generated by a, b intersects Sing g∗0 only at 0, i.e.SC0

⋂Sing g∗ = {0}. q.e.d.

5.7. Example Let g0 = so(3), g = sl(2,C) ∼= C3. Then Sing g∗ = {0},C = {z ∈ g∗; zlinearly independent withz}; consequently C is describedby two real equations: z1z2 − z2z1 = 0, z1z3 − z3z1 = 0. The set {z ∈g∗; zlinearly independent withz} is contained in C for arbitrary g.

5.8. Now, we shall introduce a remarkable pair of complex bivectorson g∗ playing the crucial role in the sequel of the paper. This pair is(c, c), where c is as in 5.1 and c is given by c = ck

ij zk∂

∂zi∧ ∂

∂zj. One can

define c intrinsically by the diagram

g∗ c−→ g∗∧

g∗

↑ · ↑ cg∗ = g∗,

where c is from (5.1.1) and · stands for the complex conjugation corre-sponding to the real form g0 ⊂ g.

26

5.9. Proposition (i) c is G0-invariant;

(ii) (c, c) is a complex Poisson pair;

(iii) (c, c) is complete at any point z ∈ g∗ \ C (see Definition 2.17).

Proof. (i), (ii) are obtained by direct calculations. The last assertionfollows from Theorem 5.6 since the set C consits precisely of the pointsof incompleteness for (c, c). Indeed, rank(λ1c + λ2c)(z) = rank ck

ij(zk +zk) ∂

∂zi∧ ∂

∂zjis less than maximal if and only if λ1z + λ2z ∈ Sing g∗.

q.e.d.The end of this section is devoted to the study of the first integrals

(Definition 2.12) for the bihamiltonian structure (J, c, c).

5.10. Definition Let r = corank c (codimension of symplectic leaf ofmaximal dimension). Let us write rank g for r and call this number therank of g.

Note that for the semisimple case this notion of rank coincides withthe standard one, i.e. with dimension of a Cartan subalgebra.

5.11. Definition Let Zholc (U) denote the space of holomorphic Casimir

functions for c over an open set U ⊂ g∗.An open set U ⊂ g∗ \ Sing g∗ is called admissible if there exist r =

rank g functionally independent functions from Zholc (U).

5.12. Proposition Let a set U be admissible. Given a function g ∈Zhol

c (U), define a function g ∈ EC(U) by the formula g = ( ∂g∂zi

)zi. Thenthe space Zc(U) of (smooth) Casimir functions for c is equal to {g; g ∈Zhol

c (U)}∪O(U), where O(U) is the space of antiholomorphic functionsover U .

Proof. The following calculation shows that g ∈ Zc(U):

c(g)j = ckij zk

∂g

∂zi= ck

ij zk(∂g

∂zi) = c(g)j = 0

(here vj stands for j-th component of a vector field v = vi∂

∂zi).

Now, let g1, . . . , gr ∈ Zholc (U) be functionally independent. We note

that the (1, 0)-differentials ∂g1, . . . , ∂gr are linearly independent pre-cisely at those points where ∂g1, . . . , ∂gr are. Thus by the dimensionarguments (rank c = rank c) the functions g1, . . . , gr together with theantiholomorfic functions functionally generate the space Zc(U). q.e.d.

27

5.13. Definition Define φλ : g∗ → g∗, λ = (λ1, λ2) ∈ C2 \ {(0, 0)} byφλ(z) = λ1z +λ2z. This is a R-linear isomorphism if |λ1| 6= |λ2| and anepimorphism onto an n-dimensional (n = dimC g) subspace otherwise.

An open set U ⊂ g∗ \ C is called λ-admissible if the set φλ(U) hasan admissible neighbourhood.

An open set U ⊂ g∗ \ C is called strongly admissible if it is λ-admissible for any λ ∈ C2 \ {(0, 0)}.

5.14. Proposition Let a set U ⊂ g∗ be λ-admissible and let Uλ be anadmissible neighbourhood of φλ(U).

Then the space Zcλ(U) of (smooth) Casimir functions for cλ = λ1c+

λ2c, λ1 6= 0, is equal to {g ◦ φλ|U ; g ∈ Zholc (Uλ)} ∪ O(U).

Proof. Again, let g1, . . . , gr ∈ Zholc (Uλ) be functionally independent.

Obviously, the functions gλ,1 = g1 ◦ φλ, . . . , gλ,r = gr ◦ φλ are Casimirfunctions for cλ. They are functionally independent on U since theJacobi matrices D = ∂(g1,...,gr)

∂(z1,...,zn) and Dλ = ∂(gλ,1,...,gλ,r)∂(z1,...,zn) are related as

followsDλ(z) = λ1D ◦ φλ(z).

So gλ,1, . . . , gλ,r and O(U) generate Zcλ(U). q.e.d.

The following proposition shows that strongly admissible sets existand describes them in the semisimple case.

5.15. Proposition (i) Let || · || be a norm in g∗ Then any open setU with a sufficiently small diameter diamU = supz,z′∈U ||z − z′||is strongly admissible.

(ii) Assume that g is semisimple. Then any open set U ⊂ g∗ \ C isstrongly admissible.

Proof. (i) We start from the following claim: if a set U is admissible,then the set λU = {λu; u ∈ U} is so for any λ ∈ C \ {0}. Indeed, thebivector c is homogeneous with homogeneity degree 1: hλ,∗c = λc, wherehλ(z) = λz. Hence, if g1, . . . , gr are independent Casimir functions forc over U , then (hla

∗)−1g1, . . . , (h∗λ)−1gn are so over hλ(U) = λU .Now, assume that the norm is so chosen that ||z|| = ||z||. Then the

inequality

||λ1z + λ2z − λ1z′ − λ2z

′|| ≤ |λ1| ||z − z′||+ |λ2| ||z − z′||

28

shows thatdiamφλ(U) ≤ (|λ1|+ |λ2|) diamU, (5.15.1)

where φλ is from Definition 5.13.Next, choose a point z ∈ U such that z is linearly independent with

z and consider the map C2 3 λ 7→ φλ(z) ∈ g∗. The image of the unitsphere S1 = {|λ1|2 + |λ2|2 = 1} under this map can be covered by afinite number of admissible balls B1, . . . , Bm. Inequality 5.15.1 showsthat shrinking U if needed one can get the following

φλ(U) ⊂ ∪mi=1Bi ∀λ ∈ S1.

Hence for sufficiently small U 3 x the set φλ(U), where λ ∈ S1, possessesan admissible neighbourhood and by the above proved claim the sameis true for φλ(U), λ 6= 0. Since all norms on g∗ are equivalent thiscompletes the proof.(ii) It is enough to note that there exists a set g1(z), . . . , gr(z), r =rank g of global holomorphic Casimir functions for c that are functionallyindependent on g∗ \ Sing g∗. One can identify g and g∗ by means of theKilling form and take for g1, . . . , gr an algebraic basis of the ring ofG-invarinant polynomials on g. The functional independence of thesefunctions on g∗ \ Sing g∗ is established in Theorem 0.1 of [15]. q.e.d.

We summarize the above results in the following Proposition.

5.16. Proposition Let U ⊂ g∗ be a strongly admissible set and letUλ be an admissible neighbourhood of φλ(U). The set of first integrals(see Definition 2.12) F0(U) of (J, c, c) over U is generated by the setsZhol

c (U), {g; g ∈ Zholc (U)}, {g ◦ φλ|U ; g ∈ Zhol

c (Uλ)},O(U), where λ =(λ1, λ2) ∈ C2, λ1, λ2 6= 0.

Proof follows from Propositions 5.12,5.14 and from the definition ofthe set F0(U). q.e.d.

The following proposition will be crucial in the proof of our mainresult (Theorem 6.7). As usual, given a subspace V ⊂ (T Cz g∗)∗, we setV 1,0 = V ∩ (T 1,0

z g∗)∗.

5.17. Proposition Let

µ(z) = dim(ker c(z))1,0 ∩ (ker c(z))1,0

and letµλ(z) = dim(ker c(z))1,0 ∩ (ker cλ(z))1,0

for cλ = λ1c + λ2c, λ = (λ1, λ2) ∈ C2, λ1, λ2 6= 0.Then

29

(i) µ(z) = µλ(z) for any z ∈ g∗ \ C;(ii) there exists a real algebraic set R, C ⊂ R ⊂ g∗, where C is the

incompleteness set (see Definition 5.3), such that µ(z) = µ isconstant and minimal on g∗ \ R and µ(z) > µ for z ∈ R;

(iii) if g is semisimple the set R \ C is empty and µ(z) ≡ 0 on g∗ \ C.

Proof. (i) We shall use the completeness of the bihamiltonian struc-ture (J, c, c) at any z ∈ g∗ \ C (see proof of Proposition 5.9). By The-orem 2.20 and Corollary 2.21 the pair c(z), c(z) ∈ ∧2 T 1,0

z g∗ does nothave the Jordan blocks in its decomposition. Thus we can use Proposi-tion 2.24 to deduce that µ(z) = µλ(z).(ii) To prove this condition it is sufficient to note that the subspace(ker c(z))1,0∩(ker c(z))1,0 annihilates the sum of characteristic subspacesPc,z + Pc,z. Put R = {z ∈ g∗; dim(Pc,z + Pc,z) < m}, where m =maxz dim(Pc,z + Pc,z). The defining polynomials for R are the minorsof m-th order of the 2n× n-matrix

∣∣∣∣∣

∣∣∣∣∣ckijzk

ckij zk

∣∣∣∣∣

∣∣∣∣∣ .

If the above defined set R lies in C, let us change the definition andput R = C.

It remains to prove the inclusion C ⊂ R in the case C 6⊃ R, C 6= R.Introduce a set Rλ = {z ∈ g∗; dim(Pc,z + Pcλ,z) < mλ}, where mλ =maxz dim(Pc,z +Pcλ,z). Then by (i) Rλ, λ2 6= 0 coincides with R outsideC. Since R = Cl(R \ C) and Rλ = Cl(Rλ \ C) (Zarisski closures), onegets R = Rλ.

Now, let z ∈ C and let λ = (λ1, λ2) ∈ C2, λ2 6= 0 be such thatrank(λ1c + λ2c)(z) < R0 (cf. the definition of completeness, 2.17).Then dim(Pc,z + Pcλ,z) < mλ. Consequently, C ⊂ Rλ = R. If the only(up to the proportionality) bivector of nonmaximal rank in the family{cλ(z)} is c, then dim(Pc,z + Pc,z) < m and again C ⊂ Rλ = R.(iii) It follows from the proof of Proposition 2.24 that dim(ker c(z))1,0∩(ker c(z))1,0 is equal to dimension of the sum of the trivial Kroneckerblocks for the pair c(z), c(z). If we consider c, c as elements of Γ(

∧2 T 1,0g∗),the same arguments as for the method of argument translation ( [17],Theorem 4.1) show that in the semisimple case the trivial Kroneckerblocks are absent for any z ∈ g∗ \ C. q.e.d.

30

5.18. Definition We call the set R from Proposition 5.17 the Kro-necker irregularity set and the number µ the trivial Kronecker dimensionof the Lie algebra g.

The following example shows that for nonsemisimple Lie algebrasthe set R \ C may be nonempty and the trivial Kronecker dimensionmay be nonzero.

5.19. Example Let g = Span{p1, p2, q1, . . . , q4, f1, . . . , f4, g1, . . . , g4}be a fourteen-dimensional Lie algebra with the standard linear Poissonbivector c = ∂

∂p1∧ (f1

∂∂q1

+ · · · + f4∂

∂q4) + ∂

∂p2∧ (g1

∂∂q1

+ · · · + g4∂

∂q4).

Then R is given by one real equation∣∣∣∣∣∣∣∣∣

f1 f2 f3 f4

g1 g2 g3 g4

f1 f2 f3 f4

g1 g2 g3 g4

∣∣∣∣∣∣∣∣∣= 0.

Also, Sing g∗ consists of the points where the vectors f1∂

∂q1+ · · · +

f4∂

∂q4, g1

∂∂q1

+ · · ·+ g4∂

∂q4are linearly dependent, i.e. the defining equa-

tions for Sing g∗ are f1g2 − f2g1 = 0, f1g3 − f3g1 = 0, f1g4 − f4g1 =0. However, the first part of the proof of Proposition 5.6 shows thatcodimR C ≥ codimR Sing g∗−4; consequently, in our example codimR C ≥6− 4 = 2 and C 6= R.

Here µ = 0Adding one more dimension to g and retaining c as above one gets

µ = 1. Also, µ will be nonzero for all reductive nonsemisimple Liealgebras.

Note that the above examples agree with our Convention 5.3.

6 CR-geometry of real coadjoint orbits

We retain the notations and conventions from the previous section.

6.1. Proposition (i) The bivectors c1 = Re c, c2 = Im c are Poisson.

(ii) The coadjoint action of GR, where GR is G considered as a real Liegroup, is hamiltonian with respect to c1, c2 (see Definition 1.21).

(iii) The generalized distribution of subspaces tangent to the G0-orbitsis generated by the vector fields c(zi) + c(zi), i = 1, . . . , n.

31

Proof. (i) The more general statement that the pair c1, c2 is Poissonis proved by the same arguments as in Example 2.7.(ii) First, we shall prove that the holomorphic coadjoint action of G ong∗ is hamiltonian in holomorphic sense with respect to c. Consider theantirepresentation Ad∗ : G → g∗. The corresponding Lie algebra actionρ : g → Vecthol(g∗), where Vecthol(g∗) is the Lie algebra of holomorphicvector fields on g∗, can be described as follows. The vector field ρ(v), v ∈g is equal to

z 7→ ad∗(v)z : g∗ → g∗ ∼= T 1,0z g∗, v ∈ g.

On the other hand, if e1, . . . , en, ckij are as in 5.1, then

< ad∗(ei)z, ej >=< z, ad(ei)ej >=< z, [ei, ej ] >=< z, ckijek >= ck

ijzk,

where we put z = z1e1 + · · · + znen. Hence, ρ(ei) = ck

ijzk∂

∂zj= c(zi)

and the corresponding antihomomorphism ψ : g → O(g∗) (cf. Defini-tion 1.21) is defined by ei 7→ zi, i = 1, . . . , n.

Now, the Lie algebra action ρR : gR ∼= g0 ⊕ ig0 → Vect(g∗) corre-sponding to the antirepresentation Ad∗ : GR → g∗ is described by theformulas

< ad∗(ei)z, ej >= ckijxk, < ad∗(iei)z, ej >= ck

ijyk

< ad∗(ei)z, iej >= ckijyk, < ad∗(iei)z, iej >= −ck

ijxk.

Consequently,

ρR(ei) = ckijxk

∂

∂xj+ ck

ijyk∂

∂yj= (c + c)(zi + zi) = (c− c)(zi − zi)

and

ρR(iei) = ckijyk

∂

∂xj− ck

ijxk∂

∂yj= (c + c)(zi − zi) = (c− c)(zi + zi).

(iii) This condition follows from the proof of (ii) and from the obviousequality (c + c)(zi + zi) = c(zi) + c(zi). q.e.d.

6.2. Proposition Let O be a G0-orbit through z0 ∈ g∗. Then O is ageneric CR-manifold in the G-orbit G(z0);

32

Proof. The G0-invariance of the complex sructure J on g∗ implies theconstancy of dimTzO

⋂J TzO, z ∈ O. To prove the genericity we notethat the tangent bundle TO is generated by the vector fields c(zj) +c(zj), j = 1, . . . , n (Proposition 6.1,(iii)), and that J acts on them asfollows

J (c(zj) + c(zj)) = i(c(zj)− c(zj)). (6.2.1)

Thus TO+J TO is generated by the real and imaginary parts of the vec-tor fields c(zj), j = 1, . . . , n, spanning T 1,0G(z0). Hence TzO+J TzO =T Cz G(z0). q.e.d.

The next proposition describes describes CR-dimension of a G0-orbit. This result will not be used in the sequel.

6.3. Proposition Let O be a G0-orbit through z0 ∈ g∗. Write Gz (Gz0)

for the stabilizer of z ∈ g∗ in G (G0) and gz (gz0) for the corresponding

Lie algebra. ThenT 1,0

z O ∼= gz/(gz0)C ∀z ∈ O.

Proof. We study the embedding of TzO in the tangent space TzG(z0)to a G-orbit G(z0). At the Lie algebra level it is equal to a map

ι : g0/gz0 −→ g/gz

induced by the inclusions g0 ↪→ g, gz0 ↪→ gz. The intersection ι(g0/gz

0) ∩J ι(g0/gz

0), where J is the complex structure on g/gz induced by themultiplication by i, is equal to

{s + gz0; s ∈ g0,∃t ∈ g0, s− it ∈ gz}.

We define a map

φ : gz −→ ι(g0/gz0) ∩ J ι(g0/gz

0) ∼= TzO⋂J TzO

by the formulagz 3 s− it 7→ s + gz,

where s, t ∈ g0. The kernel of φ is equal to {s − it ∈ gz; s ∈ gz} ={s − it ∈ gz; s ∈ gz

0 = gz ∩ g0}. Since ad∗s(x) = ad∗s(y) = 0 fors ∈ gz

0, x = Re z, y = Im z, one gets

0 = ad∗(s− it)(x + iy) = −iad∗(t)(x + iy) ⇒ad∗(t)x = ad∗(t)y = 0 ⇒ t ∈ gz

0.

Thus kerφ = (gz0)C. The surjectivity of φ is obvious. q.e.d.

33

6.4. Theorem Let O be a G0-orbit in a G-orbit M of maximal di-mension. Then T 1,0O is generated by the vector fields c(g1), . . . , c(gr).

Proof. By (6.2.1) TO⋂J TO is generated by linear combinationsαj(c(zj) + c(zj)), αj ∈ E(g∗), such that there exist βj ∈ E(g∗) satisfyingthe equality

αj(c(zj) + c(zj)) = βj i(c(zj)− c(zj)).

This implies(αj + iβj)c(zj) + (αj − iβj)c(zj) = 0

andγjc(zj) = 0, (6.4.1)

where we put γj = αj + iβj .In order to calculate all vector functions γ = (γ1, . . . , γn) satisfying

(6.4.1) one observes two facts. First that given g1, . . . , gr as in ??,the vector functions γm = (∂gm

∂z1, . . . , ∂gm

∂zn), i = 1, . . . , r, satisfy (6.4.1).

Second, the dimension arguments show that any γ(z) for which (6.4.1)holds is a linear combination of γ1(z), . . . , γr(z) if z ∈ g∗ \ Sing g∗.

In other words, TCRO is generated by (∂gm

∂zj+ ∂gm

∂zj)(c(zj)+c(zj)), i =

1, . . . , r, and T 1,0O by

(∂gm

∂zj+ (

∂gm

∂zj))(c(zj) + c(zj))− iJ (

∂gm

∂zj+ (

∂gm

∂zj))(c(zj) + c(zj)) =

(∂gm

∂zj+ (

∂gm

∂zj))(c(zj) + c(zj) + c(zj)− c(zj)) =

2c(gm) + 2(∂gm

∂zj)c(zj) = 2c(gm).

q.e.d.Our next aim is to study G0-orbits in g∗ from the symplectic point

of view.

6.5. Corollary In the assumptions of Theorem 6.4 O is a CR-isotropicsubmanifold in M (Definition 1.19).

Proof. First we shall show the G0-invariance of the functions g1, . . . , gr.This fact follows from the equality

clij

∂gm

∂zi+ ck

ijzk∂2gm

∂zi∂zl= 0 (6.5.1)

34

obtained by the differentiation of the equality ckij

∂gm

∂zi= 0 with respect

to zl. Next, conjugating (6.5.1) and multiplying by zl one gets

0 = clijzl(

∂gm

∂zi) + ck

ij zk(∂2gm

∂zi∂zl)zl =

(ckijzk

∂

∂zi+ ck

ij zk∂

∂zi)[(

∂gm

∂zl)zl] =

−(c(zj) + c(zj))gm.

Now, recall that (T 1,0O)⊥ω is generated by the vector fields c(f),where f runs through all G0-invarinat functions. Thus by Theorem 6.4T 1,0O ⊂ (T 1,0O)⊥ω. q.e.d.

6.6. Let R denote a sum of G0-orbits of the principal orbital type ong∗ and let R be its complement. Recall ([7]) that R is a nowhere denseclosed subset and that for any G-orbit M a factor (M \ R)/G0 is amanifold.

6.7. Theorem Let M ⊂ g∗ be a G-orbit that is not contained in C⋃Rand let p : M \R −→ (M \R)/G0 be the canonical projection. Write Jfor the symplectic bihamiltonian structure on M \R associated with therestriction of the standard holomorphic symplectic form ω = (c|M )−1.Then the reduction J ′ of J via p is a complete bihamiltonian structure onM ′ = (M \R)/G0. More precisely, J ′ is complete at any z′ 6∈ p(M

⋂ C).The realization J of J ′ is minimal (see Definition 4.6).

We postpone the proof of this theorem till Subsection 6.8 and presentsome preliminary results.

6.8. Proof of Theorem 6.7. We are going to use Theorem 4.4 andthe notations from it.

The foliation K of leaves of p is a generic CR-foliation due to Propo-sition 6.2.

By Corollaries ?? and 6.5 the number kz′ = dim(ker pC∗ )1,0z

⋂

((ker pC∗ )1,0z )⊥ω(z), z ∈ p−1(z′), equals r = rank g for any z′ ∈ M ′. We

now shall prove that the number kλz′ = dim(ker pC∗ )z

⋂(ker pC∗ )

⊥ωλ(z)z , z ∈

p−1(z′) satisfies the inequality

kλz′ ≥ kz′ (6.8.1)

35

for any z′ ∈ M ′, λ = (λ1, λ2) ∈ C2 \ Λ.For that purpose we shall use the bivector cλ = (ωλ)−1 and the fact

that (ker pC∗ )⊥ωλ is generated by the vector fields cλ(f), where f variesthrough the functions constant along ker p. We shall define r functionsgλ,1, . . . , gλ,r such that the vector fields cλ(gλ,1), . . . , cλ(gλ,r) are tangentto K and are independent at any point z ∈ M \ C. These functions are

gλ,1 = g1(λ2z + λ1z), . . . , gλ,r = gr(λ2z + λ1z),

where (λ1, λ2) ∈ C2 is such that cλ = λ1c + λ2c. They are independentat any z ∈ M \ C since the Jacobi matrices Jac = ∂(g1,...,gr)

∂(z1,...,zn) and Jacλ =∂(gλ,1,...,gλ,r)

∂(z1,...,zn) are related as follows

Jacλ(z) = λ2Jac(λ2z + λ1z).

The nondegeneracy of cλ implies the independence of the vector fieldscλ(gλ,1), . . . , cλ(gλ,r) at z ∈ M \ C. The following equalities show thatthese vector fields are tangent to K (TK is spanned by c(zi) + c(zi), i =1, . . . , n)

cλ(gλ,m) = λ1c(gλ,m) + λ2c(gλ,m) =

λ1∂gλ,m

∂zic(zi) + λ2

∂gλ,m

∂zic(zi) = λ1λ2

∂gm

∂zi|λ2z+λ1z(c(zi) + c(zi)).

Here we used the obvious identities∂gλ,m

∂zi= λ2

∂gm

∂zi|λ2z+λ1z,

∂gλ,m

∂zi= λ1

∂gm

∂zi|λ2z+λ1z.

Thus we have proved (6.8.1) that is equivalent via Theorems ??, ??to the following

rank c′λ(z′) ≤ rank c′(z′), z′ ∈ M ′ \ p(C).By the lower semi-continuity of the function f(λ1, λ2) = rank(λ1c

′ +λ2c

′), (λ1, λ2) ∈ C2, this gives

rank c′λ(z′) = rank c′(z′), z′ ∈ M ′ \ p(C).Thus we have obtained the constancy of kλ

z′ in λ and the equality kλz′ =

kz′ for z′ ∈ M ′ \ p(C). Since this number is also independent of z′,condition (ii) of Theorem 4.4 is satisfied.

The minimality of the realization J for J ′ follows from Corollary 6.5.q.e.d.

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7 Some open questions

7.1. It is possible to define a notion of a Veronese web (see Introduc-tion) for an arbitrary complete bihamiltonian structure. A manuscriptwith some ideas of the author concerning this subject is in preparation.

It would be desirable to generalize the result of I.M.Gelfand andI.S.Zakharevich claiming that Veronese webs are complete local invari-ants for bihamiltonian structures of general position ([11]).

7.2. The following question is natural (cf. [19]): given a complete bi-hamiltonian structure, does there exist its local realization in a complexsymplectic manifold?

7.3. A question of uniqueness (cf. [19]): is it true that any two minimalrealizations of a complete bihamiltonian structure are locally symplec-tomorphic?

7.4. The author hopes that the study of the above questions will clar-ify some aspects of reconstructing the bihamiltonian structure from itsVeronese web (the authors of the article [11] are themselves unsatisfiedby their construction, see [11], p. 166).

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