Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds
aa r X i v : . [ m a t h . DG ] F e b Closed orbits of Reeb fieldson compact Sasakian manifolds
Liviu Ornea , Misha Verbitsky Abstract
A contact structure on a manifold can be defined as a R > -automorphic symplectic structure on the cone of that man-ifold. The associated Reeb field is uniquely defined by thecontact structure and transversal to it. A Sasakian manifoldis a manifold with an R > -automorphic K¨ahler structure onits cone; clearly, this structure is defined on top of a contactstructure. Weinstein conjecture predicts that any compactcontact manifold admits at least one closed Reeb orbit. Weprove this conjecture for Sasakian manifolds, and show thatany 2 n +1-dimensional Sasakian manifold M admits at least n + 1 closed Reeb orbits. We also show that the number ofclosed Reeb orbits is either infinite or equal to the sum ofall Betti numbers of a K¨ahler orbifold obtained as an S -quotient of M . We obtain a similar estimate for the numberof elliptic curves on a compact Vaisman manifold. Contents Liviu Ornea is partially supported by UEFISCDI grant PCE 30/2021. Misha Verbitsky is partially supported by the Russian Academic Excellence Project’5-100”, FAPERJ E-26/202.912/2018 and CNPq - Process 313608/2017-2.
Keywords:
Reeb field, Sasakian manifold, Vaisman manifold, CR manifold, projectiveorbifold, quasi-regular, elliptic curve. . Ornea, M. Verbitsky Closed orbits of Reeb fields
Initially stated for periodic orbits of Hamiltonian flows on hypersurfaces ofcontact type in symplectic manifolds, Weinstein conjecture ([We]) can beintrinsically formulated in the following form:
Weinstein Conjecture:
On any closed contact manifold (
N, η ) the Reebfield has at least one closed orbit.For history and background on Weinstein conjecture, see the survey [P].The conjecture has been solved in the affirmative for contact hyper-surfaces in Euclidean space by C. Viterbo ([Vi]), on S , with a differentmethod, by H. Hofer ([Ho]) and, more generally, on closed 3-manifolds, byC.H. Taubes ([Ta]). Several extensions of these results are also available.But in full generality, the conjecture is still open and remains one of themost challenging problems in symplectic and contact topology.The dynamic of the Reeb flow on a contact manifold became a separatesubfield in symplectic geometry, called the Reeb dynamics ([Ge2]). One ofthe central questions of this subfield is finding an estimate for the number ofclosed Reeb orbits on contact manifolds under different geometric conditions.Most research in this direction is based on contact homology, which is aMorse-type cohomology theory expressing the Reeb orbits as fixed point ofa certain gradient acrion on an appropriate loop space. As in Morse (andFloer) theory, the number of closed Reeb orbits bounds the sum of Bettinumbers of the relevant cohomology theory.The most striking achievement in this direction is a result of Hutchinsand Cristofaro-Gardiner, who proved that any 3-dimensional compact con-tact manifolds admits at least 2 distinct closed Reeb orbits ([CH]).However, for dimension > . Ornea, M. Verbitsky Closed orbits of Reeb fields
The main result of this paper is a confirmation of Weinstein conjecturefor compact Sasakian manifolds:
Theorem 1.1:
Let M be a compact Sasakian manifold of dimension 2 n + 1.Then its Reeb field has at least n + 1 closed orbits.For the proof, we first observe that the Reeb field can be approximatedby a quasi-regular one, i.e. with a Reeb field such that its space of orbitsis a projective orbifold. We then relate the closed orbits of the initial Reebfield to the fixed points of the action of the group generated by its flowon this projective orbifold. Finally, we apply the orbifold version, [Fo], ofa celebrated result of A. Bia lynicki-Birula, [Bi], counting fixed points ofalgebraic groups.Our proof does not explicitly use the contact or Sasakian geometry, buttransfers the problem to the framework of complex geometry.From this argument we could also obtain an explicit expression for thenumber of the Reeb orbits on a Sasakian manifold associated with a pro-jective orbifold X admitting holomorphic vector fields with isolated zeros.In that case the number of Reeb orbits (for an appropriate Sasakian struc-ture) is equal to the number of fixed points of any of these vector fields.It is also equal to the sum of Betti numbers of S . This always gives thelower estimate, and gives an upper estimate when the Sasakian structure issufficiently general.The compact Sasakian manifolds are known to be closely related to Vais-man manifolds (see Section 4 for their definition and properties). Indeed,Vaisman manifolds are mapping tori of circles with fibres compact Sasakianmanifolds. Diagonal Hopf manifolds and their complex compact submani-folds are typical examples.Exploiting this relation, we are able to count the elliptic curves on com-pact Vaisman manifolds: Theorem 1.2: let V be a compact Vaisman manifold of complex dimension n . Then V contains at least n elliptic curves.This theorem is proven in Section 4. Note that this result is elementaryfor the diagonal Hopf manifolds. – 3 – . Ornea, M. Verbitsky Closed orbits of Reeb fields
We present the necessary notions concerning Sasakian geometry. For detailsand examples, see [BG]. We start by recalling the definition of CR andcontact structures. Both of these geometric structures are subjancent to theSasakian structures.
Definition 2.1:
Let M be a smooth manifold, B ⊂ T M a sub-bundle inthe tangent bundle, and I : B −→ B an endomorphism satisfying I = − √− B , ( M ) ⊂ B ⊗ C ⊂ T C M = T M ⊗ C . Supposethat [ B , , B , ] ⊂ B , . Then ( B, I ) is called a
CR-structure on M . Definition 2.2:
Let (
M, B ) be a CR manifold and Π
T M/B : T M −→ T M/B be the projection to the normal bundle of B in T M . The tensor field B ⊗ B −→ T M/B mapping vector fields
X, Y ∈ B to Π T M/B ([ X, Y ]) is called the Frobenius form of B . It is the obstruction to the integrability of thedistribution given by B . Remark 2.3:
Let S be a CR manifold, with the bundle B of codimension1, and almost complex structure I ∈ End( B ). Since the Frobenius formvanishes when both arguments are from B , and B , , it is a pairing between B , and B , . Indeed, [ B , , B , ] ⊂ B , and [ B , , B , ] ⊂ B , . Thisproves that the Frobenius form is a Hermitian form taking values in a trivialrank 1 bundle T M/B . The Frobenius form on a CR manifold (
M, B, I ) withcodim B = 1 is called the Levi form . Remark 2.4:
If, in addition, B ⊂ T M is a contact bundle, its Levi form isnon-degenerate. Therefore, it has constant rank. If it is positive or negativedefinite, the CR manifold (
M, B, I ) is called strictly pseudoconvex . Inthis case we fix the orientation on the trivial bundle
T M/B in such a waythat the Levi form is positive definite.
Example 2.5: (i) A complex manifold (
X, I ) is CR, with B = T X . Indeed [ T , X, T , X ] ⊂ T , X is equivalent to the Newlander-Nirenberg theorem.(ii) Let ( X, I ) be a complex manifold and M ⊂ X a real hypersurface.Then B := T X ∩ I ( T X ) is a distribution of dimension dim C X −
1– 4 – . Ornea, M. Verbitsky
Closed orbits of Reeb fields which gives M the structure of a CR manifold.(iii) Let X be a complex manifold and ϕ : X → R a strictly plurisubhar-monic function. Then ω := ∂∂ϕ is positive definite and all level sets M c := ϕ − ( c ) are strictly pseudoconvex CR manifolds, with Frobeniusforms ω (cid:12)(cid:12)(cid:12) Mc .The following result was proved by D. Burns, but never published byhim, see [Le]; another proof, valid also for dim M = 3, is given in [Sc]. Theorem 2.6: (D. Burns)
Suppose M is a compact, connected, strictlypseudoconvex CR manifold of dimension 2 n + 1 >
5. The full CR automor-phism group Aut(
M, B, I ) is compact unless M is globally CR equivalent to S n +1 with its standard CR structure. Remark 2.7:
CR automorphisms are also called
CR holomorphic dif-feomorphisms . A vector field whose flow consists in CR automorphismsis called a CR holomorphic vector field.
Definition 2.8:
Let S be a manifold. Then C ( S ) := S × R > is called thecone over S . The multiplicative group R > acts on C ( S ) by dilations alongthe generators: h λ ( x, t ) −→ ( x, λt ). Definition 2.9:
A differential k -form α on a cone is called automorphic if h ∗ q α = q k α . Definition 2.10: A contact manifold is a manifold S such that its cone C ( S ) is endowed with an automorphic symplectic form ω , i.e. h ∗ λ ω = λ ω .The following characterization explains the geometry of a contact man-ifold and its relation to CR geometry: Theorem 2.11:
Let S be a differentiable manifold. The following are equiv-alent:(i) S is contact.(ii) S is odd-dimensional and there exists an oriented sub-bundle of codi-mension 1, B ⊂ T S , with non-degenerate Frobenius form Λ B Φ −→ T S/B . This B is called the contact bundle .– 5 – . Ornea, M. Verbitsky Closed orbits of Reeb fields (iii) S is odd-dimensional and there exists an oriented sub-bundle of codi-mension 1, B ⊂ T S , such that for any nowhere degenerate 1-form η ∈ Λ S which annihilates B , the form η ∧ ( dη ) k is a non-degeneratevolume form (where dim S = 2 k + 1). Every such η is called a contactform . Proof:
Well known (see [MS]).
Definition 2.12:
Let S be a contact manifold with a contact form η . Thevector field R defined by the equations: R y η = 1 , R y dη = 0 . is called the Reeb or characteristic field. Definition 2.13:
The
Riemannian cone of a Riemannian manifold (
M, g M )is C ( M ) := M × R > endowed with the metric g = t g M + dt ⊗ dt , where t is the coordinate on R > . Definition 2.14: A Sasakian structure on a Riemannian manifold (
M, g M )is a K¨ahler structure ( g, I, ω ) on its Riemannian cone C ( M ) , g = t g M + dt ⊗ dt ) such that: • I is h λ - invariant, and • ω is automorphic: h ∗ λ ω = λ ω .In this case, ( C ( M ) , g, I, ω ) is called the cone of the Sasakian manifold. Remark 2.15: (i) It can be seen that ϕ = t is a K¨ahler potential for ω , that is, ω = dd c ϕ ,where d c = IdI − . The manifold M can be identified with a level of ϕ .The contact bundle on M can be obtained as T M ∩ I ( T M ), where
T M is considered as a sub-bundle of
T C ( M ), and I the complex structureon T C ( M ). This gives a CR-structure on M . An easy calculationimplies that the restriction of dd c ϕ to the contact bundle is equal tothe Levi form. Therefore, the CR-structure on the Sasakian manifoldsis strictly pseudoconvex. – 6 – . Ornea, M. Verbitsky Closed orbits of Reeb fields (ii) Let ξ := t tdt be the Euler field on C ( M ). The complex structureof the cone is h λ invariant, thus the Euler field acts on the cone byholomorphic homotheties.(iii) Since ω is, in particular, a symplectic form, a Sasakian manifold is,in particular, a contact manifold . Its contact form is defined asfollows. Let η := ξ y ω . Then, by Cartan’s formula, dη = Lie ξ ω = ω ,and hence ( dη ) n ∧ η = n +1 ξ y ω n +1 is non-degenerate, thus each slice M × { t } ⊂ C ( M ) is contact, with contact form the restriction of η .We may consider η as a 1-form on M .(iv) The above CR structure is given by the distribution B := ker η .(v) Let R := Iξ . Clearly, R is tangent to M and transverse to the CRdistribution. One then verifies that η ( R ) = 1 and R y dη = 0, andhence R is the Reeb field of the contact manifold (
M, η ). Note that ona Sasakian manifold, the Reeb field is the g M -dual of the contact form: R = η ♯ . Moreover, its flow consists of contact isometries: Lie R g M = 0,Lie R η = 0.The relation between CR structures and Sasakian structures is clarifiedin the following result: Theorem 2.16: ([OV2])
Let (
B, I ) be a CR structure on an odd-dimensionalcompact manifold M . Then there exists a compatible Sasakian metric on M if and only if M admits a CR-holomorphic vector field which is transversalto B . Moreover, for every such field v , there exists a unique Sasakian metricsuch that v or − v is its Reeb field. Example 2.17:
Odd-dimensional spheres with the round metric are equippedwith a natural Sasakian structure, since their cones are C n \ √− dz i ∧ dz i is automorphic. Definition 2.18:
A Sasakian manifold is called quasi-regular if all orbitsof its Reeb field are compact.If M is a compact quasi-regular sSsakian manifold, with Reeb field R ,one can consider the space of orbits X := M/R which is the same as C ( M ) / h ξ, R i . In this case C ( M ) is the total space of a principal holo-morphic C ∗ -bundle over X and the corresponding line bundle has positivecurvature dd c log ϕ . Therefore, Kodaira’s theorem implies ([OV1]):– 7 – . Ornea, M. Verbitsky Closed orbits of Reeb fields
Theorem 2.19:
Let M be a compact, quasi-regular Sasakian manifold, withReeb field R . Then the space of orbits X := M/R is a projective orbifold.
Example 2.20:
1. Let X ⊂ C P n be a complex submanifold, and C ( X ) ⊂ C n +1 \ C ( X ) is obviously K¨ahler and itsK¨ahler form is automorphic, hence the intersection C ( X ) ∩ S n − isSasakian. This intersection is an S - bundle over X . This constructiongives many interesting contact manifolds, including Milnor’s exotic 7-spheres, which happen to be Sasakian.2. In general, every link of homogeneous singularity is Sasakian. Allquasi-regular Sasakian manifolds are obtained this way.3. All 3-dimensional Sasakian manifolds are quasi-regular, see [Be, Ge1].The following result shows that on a compact manifold, a Sasakian struc-ture can be aproximated with quasi-regular ones. Theorem 2.21: ([OV1])
Let M be a compact Sasakian manifold withReeb field R . Then R = lim i R i , where R i are the Reeb fields of quasi-regular Sasakian structures on M . Proof:
We present a proof slightly different from the original one, with-out making use of Theorem 2.6. Let (
B, I ) be the subjacent CR structure,which is strictly pseudoconvex, Remark 2.15, (i). Let G be the closure of theLie subgroup generated by the flow of R in the group of CR automorphismsAut( M, B, I ). Since R acts on M by isometries, G is compact. Being alsocommutative, G is a torus. Now, any vector field R ′ sitting in the Lie alge-bra of G sufficiently close to R will still be transversal to B , and hence itwill be the Reeb field of another Sasakian structure (Theorem 2.16).But a Reeb field is quasi-regular if and only if it generates a compactsubgroup, i.e. it is rational with respect to the rational structure of the Liealgebra of G . However, rational points are dense in this Lie algebra. Remark 2.22:
Note that during the above approximation process, the CRstructure remains unchanged. In particular, for each quasi-regular Reebfield R i approximating R , the complex structure on the projective manifold M/R i is the same, and corresponds to the one on the contact distributionon M . – 8 – . Ornea, M. Verbitsky Closed orbits of Reeb fields
We can prove now our main result, Theorem 1.1.Let M be a (2 n + 1)-dimensional Sasakian manifold, with Reeb field R and subjacent CR structure ( B, I ). As in the proof of Theorem 2.21, denotewith G the closure of the group generated by the flow of R in the group ofCR automorphisms Aut( M, B, I ). i.e. G := h e tR i , for t ∈ R . The followingstatement is then clear: Claim 3.1:
There exists a one-to-one correspondence between 1-dimensionalorbits of G and closed orbits of the Reeb field R .Let R ′ be a quasi-regular approximation of R (see Theorem 2.21) and X = M/R ′ the projective quotient orbifold. The key observation is: Remark 3.2:
The group G acts on X by holomorphic isometries. Moreover,there is a one-to-one correspondence between 1-dimensional orbits of theaction of G on M and fixed points of the action of G on X .We thus reduced the problem of counting closed orbits of the Reeb fieldon M to counting fixed points of a group acting by holomorphic isometrieson a projective orbifold. This, in turn, is equal to the number of zeros ofa generic vector field r ∈ Lie( G ). For compact projective manifolds, thisnumber is computed by a celebrated theorem of A. Bia lynicki-Birula, [Bi].The orbifold version that we need is due to E. Fontanari: Theorem 3.3: ([Fo]) Let r be a holomorphic vector field on a compactprojective orbifold X of complex dimension k . Then the number of zeros of r is equal to P ki =0 b i ( X ), the sum of all Betti numbers of X .Note that by Lefschetz theorem, the sum of all Betti numbers is at leastdim C X + 1, and in our case dim C X = n , which completes the proof ofTheorem 1.1. Remark 3.4:
Note that in the number of zeros exhibited in Bia lynicki-Birula’s and Fontanari’s theorems do not count multiplicities, that is, thereexist at least n + 1 distinct closed orbits.– 9 – . Ornea, M. Verbitsky Closed orbits of Reeb fields
Vaisman manifolds are a significant and much studied subclass of locallyconformally K¨ahler manifolds, see [DO]. Here we give a definition which issuitable for our purpose:
Definition 4.1:
Let (
V, I, g V ) be a Hermitian manifold such that the fun-damental form ω V ( · , · ) = g V ( · , I · ) satisfies the equation dω V = θ ∧ ω V for a ∇ g V -parallel 1-form θ , where ∇ g V is the Levi-Civita connection of g V . Then( V, I, g V ) is a Vaisman manifold and θ is its Lee form . Remark 4.2:
By [OV3, Subsection 1.3], all compact Vaisman manifolds,considered as complex manifolds, are obtained in the following way. Let C ( M ) be the K¨ahler the cone of a compact Sasakian manifold and q anon-trivial holomorphic homothety of C ( M ). Then the compact complexmanifold C ( M ) / h q i is Vaisman. Example 4.3:
Diagonal Hopf manifolds ( C n \ / h A i where A ∈ GL( n, C ) isdiagonalizable, with eigenvalues of absolute value strictly greater than 1, areVaisman. All compact submanifolds of a Vaisman manifold are Vaisman.Non-K¨ahler elliptic surfaces are Vaisman ([Be]). Remark 4.4: (i) The 2-form ω := dd c log t on C ( M ) = M × R > is q -invariant. More-over, one can see that ω = t ( ω V − dt ∧ I ( dt )), hence ω is postive-definite in the directions transversal to (cid:10) ddt , I (cid:0) ddt (cid:1)(cid:11) . and zero on (cid:10) ddt , I (cid:0) ddt (cid:1)(cid:11) .(ii) The 1-form d log t , which lives on the cone, is already q -invariant.Therefore, the 2-form ω descends to an exact form on the Vaismanmanifold V . Definition 4.5:
The foliation Σ := ker ω on V is called the canonicalfoliation of the Vaisman manifold. Remark 4.6:
The foliation Σ is generated by the g V -duals of θ and Iθ ,which are commuting, Killing and real holomorphic vector fields.The name of this foliation is motivated by the following:– 10 – . Ornea, M. Verbitsky Closed orbits of Reeb fields
Proposition 4.7:
The foliation Σ is independent on the choice of the cone C ( M ) and of the choice of the holomorphic homothety q . Proof:
Suppose we have two different exact and (semi-)positive forms ω and ω ′ . Then the sum ω := ω + ω ′ is still exact and (semi-)positive. Ifker ω = ker ω ′ , the form ω is strictly positive, which is impossible because ω and ω ′ are exact and then Stokes theorem implies R V ω dim C V = 0. Complex curves are very particular on compact Vaisman manifolds: theyhave to be elliptic, as shown by the next result.
Theorem 4.8:
Let C be a complex curve on a compact Vaisman manifold V . Then C is a leaf of the canonical foliation. In particular, C is an ellipticcurve. Proof:
As above, R C ω = 0, by Stokes, hence C is tangent to Σ =ker ω . But all compact leaves of Σ are elliptic since the tangent bundle T Σis trivial by construction.
Remark 4.9:
We proved in [OV1] that a compact Vaisman manifold V canbe holomorphically embedded in a diagonal Hopf manifold H (see Example 4.3).Intersecting V with two complementary flags of Hopf submanifolds (whichexist due to a result of Ma. Kato, [Ka]), we see that V contains at least twoelliptic curves. In fact, there are many more elliptic curves, as follows fromCorollary 4.10 below.Let now V = C ( M ) / h q i where q = h λ , the dilation along the generatorsof the cone, for some fixed λ >
1. The canonical foliation Σ on V willcorrespond to the vector fields ξ and R = Iξ of the cone. In particular,a leaf of Σ will be compact if and only if the corresponding orbit of R isclosed. Since R is the Reeb field on M , we have proven the following: Corollary 4.10:
Let M be a Sasakian manifold and V = C ( M ) / h q i thecompact Vaisman manifold corresponding to q = h λ with λ >
1. Then thenumber of closed Reeb orbits is equal to the number of elliptic curves in V .This proves Theorem 1.2. – 11 – . Ornea, M. Verbitsky Closed orbits of Reeb fields
References [Be] F.A. Belgun,
On the metric structure of non-K¨ahler complex surfaces , Math.Ann. (2000), 1–40. (Cited on pages 8 and 10.)[Bi] A. Bia lynicki-Birula,
Some theorems on actions of algebraic groups , Ann. ofMath. (2) (1973), 480–497. (Cited on pages 3 and 9.)[BG] C. Boyer, K. Galicki, Sasakian geometry,
Oxford Mathematical Monographs.Oxford Univ. Press, Oxford, 2008. (Cited on pages 2 and 4.)[CH] D. Cristofaro-Gardiner, M. Hutchings,
From one Reeb orbit to two , J. Diff.Geom. (1) (2016), 25–36. (Cited on page 2.)[DO] S. Dragomir, L. Ornea, Locally conformally K¨ahler manifolds, Progress inMath. , Birkh¨auser, 1998. (Cited on page 10.)[Fo] C. Fontanari, Towards the cohomology of moduli spaces of higher genus stablemaps , Arch. der Math. (2007), 530–535. (Cited on pages 3 and 9.)[Ge1] H. Geiges, Normal contact structures on 3-manifolds , Tˆohoku Math. J. (1997), 415–422. (Cited on page 8.)[Ge2] H. Geiges, Controlled Reeb dynamics-three lectures not in Cala Gonone ,Complex Manifolds 6 (2019), no. 1, 118-137. (Cited on page 2.)[Ho] H. Hofer,
Pseudoholomorphic curves in symplectizations with applications tothe Weinstein conjecture in dimension three , Invent. Math. (1993), 515–563. (Cited on page 2.)[Ka] Ma. Kato,
Some Remarks on Subvarieties of Hopf Manifolds , Tokyo J. Math. , Nr. 1 (1979), 47–61. (Cited on page 11.)[Le] J.M. Lee, CR manifolds with noncompact connected automorphism groups , J.Geom. Analysis, (1996), 79–90. (Cited on page 5.)[MS] Dusa McDuff, Dietmar Salamon, Introduction to Symplectic Topology , OxfordUniversity Press, 2017 (Cited on page 6.)[OV1] L. Ornea, M. Verbitsky,
An immersion theorem for compact Vaisman man-ifolds , Math. Ann. (2005), 121–143. (Cited on pages 7, 8, and 11.)[OV2] L. Ornea, M. Verbitsky,
Sasakian structures on CR-manifolds , Geom. Ded-icata, (2007), 159–173. (Cited on page 7.)[OV3] L. Ornea, M. Verbitsky,
LCK rank of locally conformally K¨ahler manifoldswith potential , J. Geom. Phys. (2016), 92–98. (Cited on page 10.)[P] F. Pasquotto,
A Short History of the Weinstein Conjecture , Jahresber. Dtsch.Math.-Ver. (2012), 119–130. (Cited on page 2.) – 12 – . Ornea, M. Verbitsky
Closed orbits of Reeb fields [Sc] R. Schoen,
On the conformal and CR automorphism groups , Geom. Funct.Anal. (1995), no. 2, 464–481. (Cited on page 5.)[Ta] C.H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture ,Geom. Topol. (2007), 2117–2202. (Cited on page 2.)[Vi] C. Viterbo, A proof of Weinstein’s conjecture in R n , Ann. Inst. HenriPoincar´e, Anal. Non Lin´eaire (1987), 337–356. (Cited on page 2.)[We] A. Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems , J.Differ. Equations (1979), 353–358. (Cited on page 2.) Liviu OrneaUniversity of Bucharest, Faculty of Mathematics,14 Academiei str., 70109 Bucharest, Romania , and:
Institute of Mathematics ”Simion Stoilow” of the Romanian Academy,21, Calea Grivitei Str. 010702-Bucharest, Romania [email protected], [email protected]
Misha VerbitskyInstituto Nacional de Matem´atica Pura e Aplicada (IMPA)Estrada Dona Castorina, 110Jardim Botˆanico, CEP 22460-320Rio de Janeiro, RJ - Brasilalso:Laboratory of Algebraic Geometry,Faculty of Mathematics, National Research University Higher Schoolof Economics, 7 Vavilova Str. Moscow, Russia [email protected], [email protected]@verbit.ru, [email protected]