On some vertical cohomologies of complex Finsler manifolds

Transcription

1 Stud. Univ. Babeş-Bolyai Math. 58(2013), No. 2, On some vertical cohomologies o complex Finsler maniolds Cristian Ida Abstract. In this paper we study some vertical cohomologies o complex Finsler maniolds as vertical cohomology attached to a unction and vertical Lichnerowicz cohomology. We also study a relative vertical cohomology attached to a unction associated to a holomorphic Finsler subspace. Mathematics Subject Classiication (2010): 53B40, 58A10, 58A12, 53B25. Keywords: complex Finsler maniold, holomorphic subspace, cohomology. Introduction The study o vertical cohomology o complex Finsler maniolds was initiated by Pitiş and Munteanu in [13]. The main goal o this paper is to study some other vertical cohomologies or orms o type (p, q, r, s) on complex Finsler maniolds as cohomology attached to a unction deined in [12] and Lichnerowicz cohomology studied by many authors, e.g. [3, 8, 16]. In this sense, in the irst section ollowing [1, 2, 9] and [13], we briely recall some preliminaries notions about complex Finsler maniolds and v- cohomology groups. In the second section, we deine a vertical cohomology attached to a unction or orms o type (p, q, r, s) on a compex Finsler maniold (M, F ) and we explain how this cohomology depends on the unction. In particular, we show that i the unction does not vanish, then our cohomology is isomorphic with the vertical cohomology o (M, F ). In the third section we deine and we study a vertical Lichnerowicz cohomology or orms o type (p, q, r, s) on a complex Finsler maniold (M, F ) and in the last section, we construct a relative vertical cohomology attached to a unction associated to a holomorphic Finsler subspace. The methods used here are closely related to those used by [4], [12] and [16].

2 252 Cristian Ida 1. Preliminaries 1.1. Complex Finsler maniolds Let π : T 1,0 M M be the holomorphic tangent bundle o a n-dimensional complex maniold M. Denote by (π 1 (U), (z k, η k )), k = 1,..., n the induced complex coordinates on T 1,0 M, where (U, (z k )) is a local chart domain o M. At local change charts on T 1,0 M, the transormation rules o these coordinates are given by z k = z k (z), η k = z k z j ηj, (1.1) where z k are holomorphic unctions and det( z k z ) 0. j It is well known that T 1,0 M has a natural structure o 2n-dimensional complex maniold, because the transition unctions z k z are holomorphic. j Denote by M = T 1,0 M {o}, where o is the zero section o T 1,0 M, and we consider T C M = T 1,0 M T 0,1 M the complexiied tangent bundle o the real tangent bundle T R M, where T 1,0 M and T 0,1 M = T 1,0 M are the holomorphic and antiholomorphic tangent bundles o M, respectively. Let V 1,0 M = ker π be the holomorphic vertical bundle over M and V 1,0 ( M) the module o its sections, called vector ields o v-type. A given supplementary subbundle H 1,0 M o V 1,0 M in T 1,0 M, i.e. T 1,0 M = H 1,0 M V 1,0 M (1.2) deines a complex nonlinear connection on M, briely c.n.c. and we denote by H 1,0 ( M) the module o its sections, called vector ields o h-type. By conjugation over all, we get a decomposition o the complexiied tangent bundle, namely T C M = H 1,0 M V 1,0 M H 0,1 M V 0,1 M. The elements o the conjugates are called vector ields o h-type and v-type, respectively. I N j k (z, η) are the local coeicients o the c.n.c. then the ollowing set o complex vector ields { δ δz k = z k N j k η j }, { η k }, { δ δz k = z k N j k η j }, { η k } (1.3) are called the local adapted bases o H 1,0 ( M), V 1,0 ( M), H 0,1 ( M) and V 0,1 ( M), respectively. The dual adapted bases are given by {dz k }, {δη k = dη k + N k j dz j }, {dz k }, {δη k = dη k + N k j dzj }. (1.4) Throughout this paper, we consider the abreviate notations k = δ and its conjugates δz k k = δ δz k., z k=., δ k η k k =, z. k k =, δ η k k = Let us consider M be a strongly pseudoconvex complex Finsler maniold [1], i.e. M is endowed with a complex Finsler metric F : T 1,0 M R + {0} satisying: (1) F 2 is smooth on M; (2) F (z, η) > 0 or all (z, η) M and F (z, η) = 0 i and only i η = 0;

3 On some vertical cohomologies o complex Finsler maniolds 253 (3) F (z, λη) = λ F (z, η) or all (z, η) T 1,0 M and λ C = C {0}; (4) the complex hessian (G jk ) = (.. j k (F 2 )) is positive deinite on M. Let (G mj ) be the inverse o (G jm ). According to [1] and [9], a c.n.c. on (M, F ) depending only on the complex Finsler metric F is the Chern-Finsler c.n.c., locally given by and it has an important property, namely CF N j k = Gmj k. m (F 2 ) (1.5) CF R i kj= δ k CF N i j δ j CF N i k= 0. (1.6) In the sequel we will consider the adapted rames and corames with respect to the Chern-Finsler c.n.c. and the hermitian metric structure G on M given by the Sasaki type lit o the undamental tensor G jk, locally given by 1.2. Vertical cohomology G = G jk dz j dz k + G jk δη j δη k. (1.7) According to [13], the set A( M) o complex valued dierential orms on M is given by the direct sum n A( M) = A p,q,r,s ( M), (1.8) p,q,r,s=0 where A p,q,r,s ( M) or simply A p,q,r,s is the set o all (p + q + r + s)-orms which can be non zero only when these act on p vector ields o h-type, on q vector ields o h-type, on r vector ields o v-type, and on s vector ields o v-type. The elements o A p,q,r,s are called (p, q, r, s)-orms on M. With respect to the adapted corames {dz k, dz k, δη k, δη k } o T C M a orm ϕ A p,q,r,s is locally given by ϕ = 1 p!q!r!s! ϕ I pj qk rh s dz Ip dz Jq δη Kr δη Hs, (1.9) where I p denotes the ordered p-tuple (i 1... i p ), J q the ordered q-tuple (j 1... j q ), K r the ordered r-tuple (k 1... k r ), H s the ordered s-tuple (h 1... h s ) and dz Ip = dz i1... dz ip, dz Jq = dz j1... dz jq, δη Kr = δη k1... δη kr and δη Hs = δη h1... δη hs, respectively. We notice that these orms are the (p + r, q + s) complex type and according to [13] i (M, F ) is a complex Finsler maniold endowed with the Chern-Finsler c.n.c., then by (1.6) the exterior dierential d admits the decomposition da p,q,r,s A p+1,q,r,s A p,q+1,r,s A p,q,r+1,s A p,q,r,s+1 A p+1,q+1,r 1,s A p+1,q,r 1,s+1 A p+1,q+1,r,s 1 A p,q+1,r+1,s 1 (1.10)

4 254 Cristian Ida which allows us to deine eight morphisms o complex vector spaces i we consider the dierent components, namely h : A p,q,r,s A p+1,q,r,s ; v : A p,q,r,s A p,q,r+1,s h : A p,q,r,s A p,q+1,r,s ; v : A p,q,r,s A p,q,r,s+1 1 : A p,q,r,s A p+1,q+1,r 1,s ; 2 : A p,q,r,s A p+1,q,r 1,s+1 3 : A p,q,r,s A p+1,q+1,r,s 1 ; 4 : A p,q,r,s A p,q+1,r+1,s 1 We remark that these operators and the classical operators and that appear in the decomposition d = + o the dierential on a complex maniold are related by = h + v and = h + v The conjugated vertical dierential operator v is locally given by v ϕ =. k (ϕ IpJ qk rh s )δη k dz Ip dz Jq δη Kr δη Hs, (1.11) k where the sum is ater the indices i 1... i p, j 1... j q, k 1... k r and h 1... h s, respectively. Also it satisies v (ϕ ψ) = v ϕ ψ + ( 1) deg ϕ ϕ v ψ or any ϕ A p,q,r,s and ψ A p,q,r,s. This operator has the property v 2 = 0 and in [13], a classical theory o de Rham cohomology is developed or the conjugated vertical dierential v, see also [9] pag. 89. Namely, the sequence 0 Φ p,q,r i F p,q,r0 v F p,q,r,1 v F p,q,r,2 v... v..., is a ine resolution or the shea Φ p,q,r o germs o v -closed (p, q, r, 0)-orms on M, where F p,q,r,s are the sheaves o germs o (p, q, r, s)-orms. It is also given a de Rham type theorem or the v-cohomology groups H p,q,r,s ( M) = Z p,q,r,s ( M)/B p,q,r,s ( M) o the complex Finsler maniold: H s ( M, Φ p,q,r ) H p,q,r,s ( M), where Z p,q,r,s ( M) is the space o v -closed (p, q, r, s)-orms and B p,q,r,s ( M) is the space o v -exact (p, q, r, s)-orms globally deined on M. 2. Vertical cohomology attached to a unction In this section, we consider a new v-cohomology attached to a unction on the complex Finsler maniold (M, F ). This new cohomology is also deined in terms o orms o type (p, q, r, s) on M. More precisely, i (M, F ) is a complex Finsler maniold and is a unction on M, we deine the coboundary operator v, : A p,q,r,s A p,q,r,s+1, v, ϕ = v ϕ (p + q + r + s) v ϕ. (2.1) It is easy to check that v, 2 = 0 and denote by Hp,q,r,s ( M) the cohomology groups o the dierential complex (A p,q,r, ( M), v, ), called the vertical de Rham cohomology groups attached to the unction o complex Finsler maniold (M, F ).

5 On some vertical cohomologies o complex Finsler maniolds 255 More generally, or any integer k, we deine the coboundary operator v,,k : A p,q,r,s A p,q,r,s+1, v,,k ϕ = v ϕ (p + q + r + s k) v ϕ. (2.2) We still have v,,k 2 We shall restrict our attention to the cohomology H p,q,r,s = 0 and we denote by Hp,q,r,s,k ( M) the cohomology o this complex. ( M) but most results readily generalize to the cohomology H p,q,r,s,k ( M). Using (2.1), by direct calculus we obtain Proposition 2.1. I, g F( M) then (i) v,+g = v, + v,g, v,0 = 0, v, = v, ; (ii) v,g = v,g + g v, g v, v,1 = v, v = 1 2 ( v, v, ), and (iii) v, (ϕ ψ) = v, ϕ ψ + ( 1) deg ϕ ϕ v, ψ. Dependence on the unction A natural question to ask about the cohomology H p,q,r,s ( M) is how it depends on the unction. Similar with the Proposition 3.2. rom [12], we explain this act or our vertical cohomology. We have Proposition 2.2. I h F( M) does not vanish, then the cohomology groups ( M) and H p,q,r,s ( M) are isomorphic. H p,q,r,s h Proo. For each p, q, r, s N, consider the linear isomorphism φ p,q,r,s : A p,q,r,s ( M) A p,q,r,s ( M), φ p,q,r,s ϕ (ϕ) =. (2.3) hp+q+r+s I ϕ A p,q,r,s ( M), one checks easily that φ p,q,r,s+1 ( v,h ϕ) = v, (φ p,q,r,s (ϕ)), (2.4) so φ p,q,r,s induces an isomorphism between the cohomologies H p,q,r,s ( M) and H p,q,r,s h ( M). Corollary 2.3. I the unction does not vanish, then H p,q,r,s ( M) is isomorphic to the vertical de Rham cohomology H p,q,r,s ( M). Proo. We take h = 1 in the above proposition. 3. Vertical Lichnerowicz cohomology In this section we deine a vertical Lichnerowicz cohomology or (p, q, r, s)-orms on a complex Finsler maniold (M, F ) ollowing the classical deinition, e.g. [3, 8, 16]. Let ω A 0,0,0,1 ( M) be a v -closed (0, 0, 0, 1)-orm on M and the map v,ω : A p,q,r,s ( M) A p,q,r,s+1 ( M), v,ω = v ω. (3.1) Since v ω = 0, we easily obtain that v,ω 2 = 0. The dierential complex 0 A p,q,r,0 ( M) v,ω A p,q,r,1 ( M) v,ω... v,ω A p,q,r,n ( M) 0 (3.2)

6 256 Cristian Ida is called v-lichnerowicz complex o complex Finsler maniold (M, F ); its cohomology groups Hω p,q,r,s ( M) are called v-lichnerowicz cohomology groups o the complex Finsler maniold (M, F ). This is a version adapted to our study o the classical Lichnerowicz cohomology, motivated by Lichnerowicz s work [8] or Lichnerowicz-Jacobi cohomology on Jacobi and locally conormal symplectic geometry maniolds, see [3, 7]. We also notice that Vaisman in [16] studied it under the name o adapted cohomology on locally conormal Kähler (LCK) maniolds. We notice that, locally, the v-lichnerowicz complex becames the v-complex ater a change ϕ e ϕ with a unction which satisies v = ω, namely v,ω is the unique dierential in A p,q,r,s ( M) which makes the multiplication by the smooth unction e an isomorphism o cochain v-complexes e : (A p,q,r, ( M), v,ω ) (A p,q,r, ( M), v ). Proposition 3.1. The v-lichnerowicz cohomology depends only on the v-class o ω. In act, we have H p,q,r,s ω ( M) v Hω p,q,r,s ( M). Proo. Since v,ω (e ϕ) = e v,ω v ϕ it results that the map [ϕ] [e ϕ] is an isomorphism between H p,q,r,s ω ( M) v and Hω p,q,r,s ( M). Example 3.2. Let us consider ω := γ to be the conjugated vertical Liouville 1-orm (the dual o the conjugated vertical Liouville vector ield Γ = η k ). Then, by the η k homogeneity conditions o complex Finsler metric, it is locally given by γ = G jk ηj F 2 δη k = 1 F 2 F 2 η k δηk = v (log F 2 ). (3.3) Then γ is a v -closed (0, 0, 0, 1)-orm on M and we can consider the associated v- Lichnerowicz cohomology groups H p,q,r,s γ ( M). As in the classical case, using the deinition o v,ω we easily obtain v,ω (ϕ ψ) = v ϕ ψ + ( 1) deg ϕ ϕ v,ω ψ. Also, i ω 1 and ω 2 are two v -closed (0, 0, 0, 1)-orms on M then v,ω1+ω 2 (ϕ ψ) = v,ω1 ϕ ψ + ( 1) deg ϕ ϕ v,ω2 ψ, which says that the wedge product induces the map : H p,q,r,s1 ω 1 ( M) H p,q,r,s2 ω 2 ( M) H p,q,r,s1+s2 ω 1+ω 2 ( M). Corollary 3.3. The wedge product induces the ollowing homomorphism : H p,q,r,s ω ( M) H p,q,r,s ( M) H p,q,r,2s ( M). ω Now, using an argument inspired rom [16], we prove that the v-lichnerowicz cohomology spaces Hω p,q,r,s ( M) can also be obtained as the v-cohomology spaces o M with the coeicients in the shea Φ p,q,r ω o germs o v,ω -closed (p, q, r, 0)-orms. Firstly, we notice that v,ω satisies a Dolbeault type Lemma or (p, q, r, s)-orms on M. Indeed, let ϕ be a local (p, q, r, s)-orm such that v,ω ϕ = 0. Since v ω = 0 and the lemma has to be local, we may suppose ω = ( v α)/α, where α is a nonzero

7 On some vertical cohomologies o complex Finsler maniolds 257 smooth unction on M. Then, v,ω ϕ = 0 means v (αϕ) = 0, whence by Dolbeault type lemma or the operator v, see [13], we have ϕ = v,ω (ψ/α) or some local (p, q, r, s 1)-orm ψ. This is exactly the requested result. Then, we see that 0 Φ p,q,r ω is a ine resolution o Φ p,q,r ω, which leads to i A p,r,r,0 ( M) v,ω A p,q,r,1 ( M) v,ω... (3.4) Proposition 3.4. For every v -closed (0, 0, 0, 1)-orm ω, one has the isomorphisms H s ( M, Φ p,q,r ω ) Hω p,q,r,s ( M). For every ω as above, let us consider now the auxiliary vertical operator v = v p + q + r + s ω, (3.5) 2 where (p, q, r, s) is the type o the orm acted on. We notice that v is an antiderivation o dierential orms and it is easy to see that 2 v = 1 2 ω v. Then v deines a twisted v-cohomology, [17], o (p, q, r, s)-orms on M, which is given by H p,q,r, Ker ( M) = v v Im v Ker (3.6) v and is isomorphic to the cohomology o the v-complex (Ãp,q,r, ( M), v ) consisting o the (p, q, r, s)-orms ϕ A p,q,r,s ( M) satisying v 2ϕ = ω vϕ = 0. The v-complex Ãp,q,r, ( M) admits a v-subcomplex A p,q,r, ω ( M), namely, the ideal generated by ω. On this subcomplex, v = v, which means that it is a v-subcomplex o the usual v-de Rham complex o M. Hence, one has the homomorphisms a : H s (A p,q,r, ω ( M)) H p,q,r,s ( M), b : H s (A p,q,r, ω ( M)) H p,q,r,s ( M, C). (3.7) v Now, we can easily construct a homomorphism c : H p,q,r,s v ( M) H p,q,r,s+1 ( M, C). (3.8) Indeed, i [ϕ] H p,q,r,s v ( M), where ϕ is v -closed (p, q, r, s)-orm, then we put c([ϕ]) = [ω ϕ], and this produces the homomorphism rom (3.8). We notice that the existence o c gives some relation between v and the v-cohomology o M with values in C. Remark 3.5. From (2.1) and (3.5) one gets 1 v, = v p + q + r + s v (log 2 ) = 2 v, with ω = v (log 2 ). (3.9) Then, i does not vanish, we have the homomorphisms ã : H p,q,r,s ( M) H p,q,r,s ( M). (3.10) v In particular, we can choose = F to be the complex Finsler unction, and so 1 F v,f = v with ω = γ.

8 258 Cristian Ida 4. A relative vertical cohomology attached to a unction The relative de Rham cohomology was irst deined in [4] p. 78. In this subsection we construct a similar version or our vertical cohomology o complex Finsler maniolds. For the begining we need some basic notions about holomorphic Finsler subspaces. For more details see [9, 10, 11] Holomorphic Finsler subspaces Let (M, F ) be a complex Finsler space, (z k, η k ), k = 1,..., n complex coordinates in a local chart, and i : M M a holomorphic immersion o an m-dimensional complex maniold M into M, locally given by z k = z k (ξ 1,..., ξ m ). Everywhere the indices i, j, k,... run rom 1 to n and α, β, γ,... run rom 1 to m n. Let T 1,0 M and T 1,0 M be the corresponding holomorphic tangent bundles. By i,c : T 1,0 M T 1,0 M we denote the inclusion map between the maniolds T 1,0 M and T 1,0 M (the complexiied tangent inclusion map), that is i,c (ξ, θ) = (z(ξ), η(ξ, θ)), where ξ = (ξ α ), θ = θ α ξ, η = η k. Then i α z k,c has the ollowing local representation [9]: z k = z k (ξ 1,..., ξ m ), η k = θ α B k α(ξ) where B k α(ξ) = zk ξ α. (4.1) The holomorphic immersion assumption implies that Bα k = zk = 0 and ξ α Bk α = zk ξ = α 0. In a point o the complexiied tangent space T C (T 1,0 M), the local rame { ξ α, θ α } is coupled to { z k, η k } as ollows: ξ α = Bk α z k + Bk 0α η k, where B k 0α = Bk α ξ β θ β. Its dual basis satisies the conditions θ α = Bk α η k, (4.2) dz k = B k αdξ α, dη k = B k 0αdξ α + B k αdθ α (4.3) and their conjugates. In view o (4.1) the complex Finsler unction F, with the metric tensor G jk =.. j k (F 2 ), induces a complex Finsler unction F : T 1,0 M R + {0} given by F(ξ, θ) = F (z(ξ), η(ξ, θ)) = F (z k (ξ), θ α Bα(ξ)) k with the metric tensor G αβ = BαB j kg β jk. Here G αβ =.. α β (F 2 ) and α=.. θ, α β =. By these considerations, the pair (M, F) is said to be a holomorphic subspace o the complex θ β Finsler space (M, F ). From (4.2) it is deduced that the distribution V 1,0 M, spanned locally by { θ }, α = 1,..., m, is a subdistribution o the vertical distribution V 1,0 M spanned α in any point (z(ξ), η(ξ, θ)) by { }, k = 1,..., n. We consider V 1,0 M an orthogonal η k complement, namely V 1,0 M = V 1,0 M V 1,0 M, spanned in any point by the set o normal vectors {N a = Ba k }, a = 1,..., n m, which we may assume orthonormal. η k Thereore, the unctions Ba(ξ, k θ) (and their conjugates) will satisy the conditions G jk (z(ξ), η(ξ, θ))bαb j a k = 0 and G jk (z(ξ), η(ξ, θ))bj ab k = δ b ab.

9 On some vertical cohomologies o complex Finsler maniolds 259 Let us now consider the moving rame R = {Bα(ξ)B k a(ξ, k θ)} along the complex Finsler subspace (M, F) and let R 1 = (Bk αba k )t be the inverse matrix associated to the moving rame R. Evidently, Bk α and Ba k are unctions o ξ, θ and B α k B k β = δ α β, B α k B k a = 0, B a kb k b = δ a b, B k αb α j + B k ab a j = δ k j. (4.4) Let N = (N α β (ξ, θ)) be a c.n.c. on T 1,0 M and consider its adapted basis {δ α := ξ N β α α, θ α:=. β and {dξ α, δθ α = dθ α N α β dξβ }. δ δξ α = θ α } and {δ α,. α} as well as its dual {dξ α, δθ α = dθ α N α β dξβ } The c.n.c. N on T 1,0 M is said to be induced by the c.n.c. N on T 1,0 M i δθ α = B α k δηk. This condition implies [9], N α β = Bα k (Bk 0β + N k j Bj β ). Proposition 4.1. ([9, 10]). The adapted bases are tied by δ α = Bαδ k k + BaH k α a.. k, α= Bα k. k, dz k = Bαdξ k α, δη k = Bαδθ k α + BaH k αdξ a α with H a α = B a j (Bj 0α + N j k Bk α). By above proposition we easily deduce that δ α = Bαδ k k + HαN a a and k=. Bk α +Bk an a. A notable result in [9] asserts that the induced c.n.c. by the Chern-Finsler c.n.c. coincides with the intrinsic Chern-Finsler c.n.c. o the holomorphic subspace (M, F) Relative vertical cohomology Now we return to the construction o a relative vertical cohomology attached to a unction o complex Finsler maniolds. Let us denote by J C i = i,c. By Proposition 4.1. we easily deduce that i ϕ A p,q,r,s ( M) is locally given by (1.9) then (J C i) ϕ A p,q,r,s ( M) A p+h,q+k,r h,s k ( M). (4.5) h=1,r;k=1,s Thus, (J C i) does not preserves the (p, q, r, s) type components o a orm ϕ A p,q,r,s ( M), but we can eliminate this inconvenient i we take p = q = m = dim C M. Then, we easily obtain Proposition 4.2. I ϕ A m,m,r,s ( M) then (J C i) ϕ A m,m,r,s ( M). Proposition 4.3. I ϕ A m,m,r,s ( M) then. α v (J C i) ϕ = (J C i) v ϕ. (4.6) Proo. Let ϕ = ϕ ImJ mk rh s dz Im dz Jm δη Kr δη Hs A m,m,r,s ( M). By Proposition 4.1. we have where (J C i) ϕ = ϕ AmB mc rd s (ξ, θ)dξ Am dξ Bm δθ Cr δθ Ds, ϕ AmB mc rd s (ξ, θ) = B Im A m B Jm B m B Kr C r B Hs D s ϕ ImJ mk rh s (z(ξ), η(ξ, θ)) (4.7)

10 260 Cristian Ida and B Im A m = Bα i1 1 (z(ξ))... Bα im m (z(ξ)) etc. Applying v rom (1.11) it results v (J C i) ϕ =. α (ϕ AmB mc rd s )δθ α dξ Am dξ Bm δθ Cr δθ Ds. (4.8) α Similarly, we have (J C i) v ϕ =. k (ϕ ImJ mk rh s )B k αδθ α B Im A m dξ Am B Jm B m dξ Bm B Kr C r δθ Cr B Hs D s δθ Ds and by (4.2) and (4.7) one gets (J C i) v ϕ =. α (ϕ AmB mc rd s )δθ α dξ Am dξ Bm δθ Cr δθ Ds α (4.9) which completes the proo. Now, i F(M) then by (4.6) one gets v,(jc i) (J C i) ϕ = (J C i) v, ϕ, or any ϕ A m,m,r,s ( M). (4.10) Indeed, or ϕ A m,m,r,s ( M) by direct calculus we have v,(jc i) ((J C i) ϕ) = (J C i) v ((J C i) ϕ) (2m + r + s) v ((J C i) ) (J C i) ϕ = (J C i) (J C i) ( v ϕ) (2m + r + s)(j C i) ( v ) (J C i) ϕ = (J C i) ( v ϕ) (J C i) ((2m + r + s) v ϕ) = (J C i) ( v, ϕ). We deine the dierential complex... v, A m,m,r,s (J C i) v, A m,m,r,s+1 (J C i) v,... where A m,m,r,s (J C i) = A m,m,r,s ( M) A m,m,r,s 1 ( M) and v, (ϕ, ψ) = ( v, ϕ, (J C i) ϕ v,(jc i) ψ). (4.11) Taking into account v, 2 = 2 v,(j C i) = 0 and (4.10) we easily veriy that v, 2 = 0. Denote the cohomology groups o this complex by H m,m,r,s (J C i). Now, i we regraduate the complex A m,m,r,s ( M) as à m,m,r,s ( M) = A m,m,r,s 1 ( M), then we obtain an exact sequence o dierential complexes 0 Ãm,m,r,s ( M) α A m,m,r,s β (J C i) A m,m,r,s ( M) 0 (4.12) with the obvious mappings α and β given by α(ψ) = (0, ψ) and β(ϕ, ψ) = ϕ, respectively. From (4.12) we have an exact sequence in cohomologies... H m,m,r,s 1 α (J C i) ( M) H m,m,r,s ((J C i) ) β H m,m,r,s 1 ( M) δ δ H m,m,r,s α ( M).... (J C i)

11 On some vertical cohomologies o complex Finsler maniolds 261 It is easily seen that δ = (J C i). Here µ denotes the corresponding map between cohomology groups. Let ϕ A m,m,r,s ( M) be a v, -closed orm, and (ϕ, ψ) A m,m,r,s (J C i). Then v, (ϕ, ψ) = (0, (J C i) ϕ v,(jc i) ψ) and by the deinition o the operator δ we have δ [ϕ] = [(J C i) ϕ v,(jc i) ψ] = [(J C i) ϕ] = (J C i) [ϕ]. Hence we inally get a long exact sequence... H m,m,r,s 1 α (J C i) ( M) H m,m,r,s ((J C i) ) β H m,m,r,s 1 ( M) (J Ci) (J C i) H m,m,r,s α (J C i) ( M).... Finally, similar to [14], we have Corollary 4.4. I (M, F) is an m-dimensional holomorphic Finsler subspace o an n-dimensional complex Finsler space (M, F ), then (i) β : H m,m,r,m+1 (J C i) H m,m,r,m+1 ( M) is an epimorphism; (ii) α : H m,m,r,n (J C i) ( M) H m,m,r,n+1 (J C i) is an epimorphism; (iii) β : H m,m,r,s (J C i) H m,m,r,s ( M) is an isomorphism or s > m + 1; (iv) α : H m,m,r,s (J C i) ( M) H m,m,r,s+1 (v) H m,m,r,s Reerences (J C i) is an isomorphism or s > n; (J C i) = 0 or s > max{m + 1, n}. [1] Abate, M., Patrizio, G., Finsler metrics - a global approach, 1591 Lectures Notes in Mathematics, Springer-Verlag, Berlin, [2] Aikou, T., On Complex Finsler Maniolds, Rep. Kagoshima Univ., 24(1991), [3] Banyaga, A., Examples o non d ω-exact locally conormal symplectic orms, Journal o Geometry, 87(2007), No. 1-2, [4] Bott, R., Tu, L.W., Dierential Forms in Algebraic Topology, Graduate Text in Math., 82, Springer-Verlag, Berlin, [5] Kobayashi, S., Negative vector bundles and complex Finsler structures, Nagoya Math. J., 57(1975), [6] Kobayashi, S., Complex Finsler vector bundles, Finsler geometry (Seattle, WA, 1995), Contemp. Math., 196, Amer. Math. Soc., Providence, RI, 1996, [7] de León, M., López, B., Marrero, J.C., Padrón, E., On the computation o the Lichnerowicz-Jacobi cohomology, J. Geom. Phys., 44(2003), [8] Lichnerowicz, A., Les variétés de Poisson et leurs algébres de Lie associées, J. Dierential Geom., 12(2)(1977), [9] Munteanu, G., Complex spaces in Finsler, Lagrange and Hamilton geometries, volume 141 o Fundamental Theories o Physics, Kluwer Academic Publishers, Dordrecht, [10] Munteanu, G., The equations o a holomorphic subspace in a complex Finsler space, Periodica Math. Hungarica, 55(1)(2007), [11] Munteanu, G., Totally geodesic holomorphic subspaces, Nonlinear Analysis, Real World Applications, 8(2007),

12 262 Cristian Ida [12] Monnier, P., A cohomology attached to a unction, Di. Geometry and Applications, 22(2005), [13] Pitiş, G., Munteanu, G., V -cohomology o complex Finsler maniolds, Studia Univ. Babeş-Bolyai Math., 43(1998), no. 3, [14] Tevdoradze, Z., Vertical cohomologies and their application to completely integrable Hamiltonian systems, Georgian Math. J., 5(5)(1998), [15] Vaisman, I., Cohomology and dierential orms, Translation editor: Samuel I. Goldberg. Pure and Applied Mathematics, 21, Marcel Dekker, Inc., New York, [16] Vaisman, I., Remarkable operators and commutation ormulas on locally conormal Kähler maniolds, Compositio Math., 40(1980), no. 3, [17] Vaisman, I., New examples o twisted cohomologies, Bolletino U.M.I., (7) 7-B(1993), Cristian Ida Transilvania University Faculty o Mathematics and Computer Sciences 50, Iuliu Maniu Street Braşov, Romania cristian.ida@unitbv.ro