A V -COHOMOLOGY WITH RESPECT TO COMPLEX LIOUVILLE DISTRIBUTION

Transcription

1 International Electronic Journal of Geometry Volume 5 No. 1 pp (2012) c IEJG A V -COHOMOLOGY WITH RESPECT TO COMPLEX LIOUVILLE DISTRIBUTION ADELINA MANEA AND CRISTIAN IDA (Communicated by Murat TOSUN ) Abstract. We define a complex Liouille distribution on the holomorphic tangent bundle of a complex Finsler manifold. Some new operators on ertical forms with respect to Liouille distribution are defined, a Dolbeault type lemma is proed and new cohomology groups are studied. 1. Introduction and preliminaries The idea of decomposing the exterior deriatie for real smooth or complex analytic foliated manifolds and the study of their cohomology is due to I. Vaisman (see [13, 14]). There, are proed some Poincaré type Lemmas with respect to some differential operators corresponding to (0, 1) foliated type or to (0, 1) mixed type for the analytic case, respectiely. Latter on, in [12] is studied a decomposition of the exterior differential for the complex type forms on complex Finsler manifolds and it is proed a Grothendieck-Dolbeault type Lemma for the conjugated ertical differential operator d which appears in the decomposition of the operator d corresponding to (0, 1) complex type. Also, there the - cohomology groups of a complex Finsler manifold are defined. Recently, in [9] the first author has studied a new type cohomology with respect to Liouille foliation on the tangent bundle of a real Finsler manifold and a de Rham type theorem is obtained. The main purpose of the present paper is to extend this cohomology theory on complex Finsler manifolds and to find new cohomology groups related to this spaces. Firstly, following [3, 4] and [8], we define the complex Liouille distribution on the holomorphic tangent bundle of a complex Finsler space and we get an adapted basis on the holomorphic ertical bundle with respect to the orthogonal splitting V ( M) = L ( M) {ξ}, where ξ is the complex Liouille ector field. Next, by analogy with [9], we consider new type of ertical forms with respect to conjugated Liouille distribution of (s, 0) type and (s 1, 1) type, respectiely, and we obtain a decomposition of the conjugated ertical differential operator d = d 1,0 +d 0,1 for ertical forms of (s, 0) type. Finally, by applying the results from [12] concerning Date: Receied: June 28, 2010 and Accepted: April 10, Mathematics Subject Classification. 53B40, 53C12, 53A12. Key words and phrases. Complex Finsler manifold, Liouille distribution, cohomology. 151

2 152 ADELINA MANEA AND CRISTIAN IDA to the operator d we proe a Grothendieck-Dolbeault type Lemma with respect to the operator d 1,0 and new cohomology groups are obtained and studied. Let M be a n dimensional complex manifold and (z k ), k = 1,..., n the complex coordinates in a local chart U. The complexified of the real tangent bundle T R M denoted by T C M splits into the direct sum of holomorphic and antiholomorphic tangent subbundles T M and T M, respectiely, namely T C M = T M T M. The total space of holomorphic tangent bundle π : T M M is in turn a 2n dimensional complex manifold with u = (z k, η k ), k = 1,..., n the induced complex coordinates in the local chart π 1 (U), where η = η k T z zm. k A complex Finsler space is a pair (M, F ), where F : T M R + {0} is a continuous function satisfying the conditions (i) G := F 2 is smooth on M := T M {o}, where o denotes the zero section of T M; (ii) F (z, η) 0, the equality holds if and only if η = 0; (iii) F (z, λη) = λ F (z, η) for any λ C, the homogeneity condition; (i) the complex hessian (G ij ) = ( 2 G ) is positiely definite, or equialently, η i η j it mean that the indicatrix I z = {η/g ij (z, η)η i η j = 1} is strongly pseudoconex for any z M. In the following we consider the notations: G i = G η i, G j = G η j, G ij = 2 G η i η j, G ij = 2 G η i η j According to [5], the strongly pseudoconex complex Finsler structure has the following properties: (1.1) G ij η i = 0 ; G i j η i = 0 ; G i η i = G ; G j η j = G etc. (1.2) G ijk η i = 0 ; G ij k η j = 0 ; G ijk η j = G ik (1.3) G ij η i = G j ; G ij η j = G i ; G ij η i η j = G Let V ( M) T ( M) be the holomorphic ertical bundle, locally spanned by { } and V ( M) be its conjugate, locally spanned by { }. A complex nonlinear connection, briefly c.n.c., on M is gien by a supplementary complex subbundle η k η k to V ( M) in T ( M), namely T ( M) = H ( M) V The horizontal subbundle H ( M) is locally spanned by { δ = N j δz k z k k η }, where N j j k (z, η) are the coefficients of the c.n.c., which obey a certain rule of change at the local charts change δ δz k = z j z k δ = z j z k η j. such that δz j performs. Obiously, we also hae that η k The pair {δ k := δ ; δz k:=. }, k = 1,..., n will be called the adapted frames k η k of the c.n.c. By conjugation an adapted frame {δ k ;. k } is obtained on T The dual adapted bases are gien by {dz k }, {δη k = dη k + Nj kdzj }, {dz k } and {δη k = dη k +N k j dzj } which span the dual bundles H ( M), V ( M), H ( M) and V ( M), respectiely. According to [1, 2, 11], a c.n.c. related only to the fundamental function of the complex Finsler space (M, F ) is almost classical now, the Chern-Finsler c.n.c., CF locally gien by N j k = Gmj G lm η l, where (G mj ) denotes the inerse of (G z k jm ).

3 A V -COHOMOLOGY WITH RESPECT TO COMPLEX LIOUVILLE DISTRIBUTION 153 Throughout this paper we consider the adapted frames and coframes with respect to this c.n.c. 2. A complex Liouille distribution It is well known that a strongly pseudoconex complex Finsler structure F defines a hermitian structure on the holomorphic ertical bundle V ( M) by (2.1) G (X, Y )(z, η) = G ij (z, η) Ẋi (z, η). Y j (z, η) for all X =Ẋi (z, η). i, Y = Y. j (z, η) j. Γ(V ( M)). An important global ertical ector field is defined by ξ = η i. i and it is called the complex Liouille ector field (or radial ertical ector field). We notice that the third equation of (1.3) says that (2.2) G = G (ξ, ξ) > 0 so ξ is an embedding of M into V Let {ξ} be the complex line bundle oer M spanned by ξ and we define the complex Liouille distribution as the complementary orthogonal distribution L ( M) to {ξ} in V ( M) with respect to G, namely V ( M) = L ( M) {ξ}. Hence, L ( M) is defined by (2.3) Γ(L ( M)) = {X Γ(V ( M)) ; G (X, ξ) = 0} Consequently, let us consider the ertical ector fields (2.4) X k =. k t k ξ, k = 1,..., n where the functions t k (z, η) are defined by the conditions (2.5) G (X k, ξ) = 0, k = 1,..., n Thus, the aboe conditions become G (. k, η j. j ) t k G (ξ, ξ) = 0, k = 1,..., n so, taking into account (1.3) and (2.1), we obtain the local expression of the functions t k in a local chart (U, (z i, η i )) (2.6) t k = G k G, k = 1,..., n If (U, (z i, η i )) is another local chart on M, then on U U ϕ, we hae t j = G jl η l G = 1 G z l zi z k ηk k z z j z l G ik = zi z j t i so we obtain the following changing rule for the ector fields from (2.4) (2.7) X j = zk z j X k, j = 1,..., n By conjugation we obtain the decomposition V C ( M) = L ( M) {ξ} L ( M) {ξ}.

4 154 ADELINA MANEA AND CRISTIAN IDA Proposition 2.1. The functions {t k }, k = 1,..., n, locally gien by (2.6) satisfies (2.8) t k η k = t k η k = 1 ; X k η k = X k η k = 0 (2.9). k (t l ) = G lk G t lt k ;. k (t l ) = G lk G t lt k (2.10) ξt k = t k ; ξt k = 0 ; η k. j (t k ) = t j ; η k (ξt k ) = 1 Proof. We hae that t k η k = G k G ηk = 1 and similarly t k η k = G k G ηk = 1, where we used (2.6) and the last two equalities from (1.1). Now, X k η k = (. k t k ξ)η k = 1 t k η k = 0 and similarly for conjugated. Thus, the relations (2.8) are proed. Similarly, by direct calculations using (1.1), (1.3) and (2.6), one gets (2.9) and (2.10). Proposition 2.2. There are the relations (2.11) [X i, X j ] = t i X j t j X i ; [X i, ξ] = X i (2.12) [X i, X j ] = 0 ; [X i, ξ] = 0 and its conjugates. Proof. Taking into account (2.10), [ i,. j]. = 0, G ij = G ji and the clasical properties.. of Lie brackets we obtain [X i, X j ] = t j i +t i j t i t j ξ + t j t i ξ = t i X j t j X i and [X i, ξ] = [ i,. ξ] + ξ(t i )ξ = δi k. k t i ξ = X i. Now, by direct calculations we hae [X i, X j ] = X j (t i )ξ X i (t j )ξ =. j (t i )ξ. i (t j )ξ and [X i, ξ] = [. i, ξ] t i [ξ, ξ] + ξ(t i )ξ = 0. = ( G j G t j ). i ( G i G t i). j = 0 The aboe proposition says that the distribution L ( M) is one integrable. By the conditions (2.5), {X 1,..., X n } are n ectors fields orthogonal to ξ, so they belong to the (n 1) dimensional distribution L It results that they are linear dependent and, from (2.8) (2.13) X n = 1 n 1 η n η i X i i=1 since the local coordinate η n is nonzero eerywhere. We hae Proposition 2.3. The system of complex ertical ector fields {X 1,..., X n 1, ξ} is a locally adapted basis on V ( M) with respect to the complex Liouille distribution. Proof. The proof is similar with the analogue result from real case (see [8]), and it consist to check that the rank of the matrix of change from the natural basis {. k}, k = 1,..., n of V ( M) to {X 1,..., X n 1, ξ} is equal to n.

5 A V -COHOMOLOGY WITH RESPECT TO COMPLEX LIOUVILLE DISTRIBUTION 155 As in the real case (see [9]), we hae the following concludent remark: Let (U, (z k, η k )) and (U, (z k, η k )) be two local charts which domains oerlap, where η k and η n are nonzero functions (in eery local chart on M there is at least one nonzero coordinate function η i ). The adapted basis in U is {X 1,..., X k 1, X k+1,..., X n, ξ}. In U U ϕ we hae (2.7) and (2.13), hence n 1 X j = ( zi z j ηi z n η n z j )X i ; X i = i=1 n j=1,j k j ( z z i η j z k η k z i )X j for all j = 1,..., n, i = 1,..., n 1. One can see that the aboe relation also imply z i z k ηi η n z n z k = n j=1,j k η j η k ( zi z j ηi η n z n z j ) By a straightforward calculation we hae that the determinant of the change matrix {X 1,..., X n 1, ξ} {X 1,..., X k 1, X k+1,..., X n, ξ} on V ( M) is equal to ( 1) n+k η k η n det( zi z j ) 3. New operators on ertical forms with respect to complex Liouille distribution According to [12], let us consider A p,q,r,s ( M) the set of all (p, q, r, s) -forms with complex alues on M locally defined by (3.1) φ = 1 p!q!r!s! φ I p J q H r K s (z, η)dz Ip δη Jq dz Hr δη Ks where I p = (i 1...i p ), J q = (j 1...j q ), H r = (h 1...h r ), K s = (k 1...k s ) and these forms can be nonzero only when they act on p ectors from Γ(H ( M)), on q ectors from Γ(V ( M)), on r ectors from Γ(H ( M)) and s ectors from Γ(V ( M)), respectiely. We also consider the conjugated ertical differential operator d : A p,q,r,s ( M) A p,q,r,s+1 ( M), locally gien by (3.2) d φ = k. k (φ Ip J q H r K s )δη k dz I p δη J q dz H r δη K s where the sum is after i 1 <... < i p, j 1 <... < j q, h 1 <... < h r and k 1 <... < k s. This operator has the property (d ) 2 = 0 and it locally satisfies a Grothendieck- Dolbeault type lemma (for details see [12] and [11] p. 88). Proposition 3.1. The (0, 0, 0, 1) ertical form ω 0 = t i δη i is globally defined and satisfies (3.3) ω 0 (ξ) = 1, ω 0 (X a ) = 0, ω 0 = d (ln G) for all a = 1,..., n 1, X a gien by (2.4) and G = F 2 is the fundamental function. Proof. In U U ϕ we hae ω 0 = t j δη j = zi z j t i z j z k δηk = t i δη i = ω 0

6 156 ADELINA MANEA AND CRISTIAN IDA We also hae δη i (ξ) = η i, for all i = 1,..., n, and taking into account the first relation of (2.8) it results ω 0 (ξ) = 1, ω 0 (X a ) = t i δη i (. a t a ξ) = t i δ i a t a t i η i = 0 where δa i denotes the Kronecker symbols. By conjugation in the relation (2.13) it results also ω 0 (X n ) = 0. Now, we hae which ends the proof. d (ln G) =. k (ln G)δη k = G k G δηk = t k δη k = ω 0 We notice that the equality ω 0 = d (ln G) shows that ω 0 is an d -exact ertical (0, 0, 0, 1) -form and the conjugated complex Liouille distribution is defined by the equation ω 0 = 0. In the following, we will consider Ω s ( M) := A 0,0,0,s ( M) A p,q,r,s ( M) the subspace of all ertical forms of (0, 0, 0, s) type on M. Definition 3.1. A ertical (0, 0, 0, s) -form φ Ω s ( M) is called a (s 1, s 2 ) -form iff for any ertical ector fields X 1,..., X s Γ(V ( M)) we hae φ(x 1,..., X s ) 0 only if s 1 arguments are in Γ(L ( M)) and s 2 arguments are in Γ({ξ}). Since {ξ} is a line distribution, we can talk only about (s 1, s 2 ) -forms with s 2 {0, 1}. We will denote the space of (s 1, s 2 ) -forms by Ω s 1,s 2 By the aboe definition, we hae the equialence (3.4) φ Ω s 1,1 ( M) φ(x 1,..., X s ) = 0, X 1,..., X s Γ(L ( M)) Proposition 3.2. Let φ be a nonzero ertical (0, 0, 0, s) -form on M. The following assertions are true (i) φ Ω s,0 ( M) iff i ξ φ = 0, where i X denotes the interior product. (ii) The ertical (0, 0, 0, s 1) -form i ξ φ is a (s 1, 0) -form. (iii) φ Ω s 1,1 ( M) implies i ξ φ 0. (i) If there is a (s 1, 0) -form α such that φ = ω 0 α then φ Ω s 1,1 Proof. (i) Let φ Ω s,0 ( M), hence φ(x 1,..., X s ) 0 only if all the arguments are in Γ(L ( M)). So i ξ φ is a (0, 0, 0, s 1) -form and (i ξ φ)(x 1,..., X s 1 ) = φ(ξ, X 1,..., X s 1 ) = 0 for eery ertical ector fields X 1,..., X s 1. So, i ξ φ = 0. Conersely, if φ is a (0, 0, 0, s) -form such that i ξ φ = 0, then φ(x 1,..., X s ) = 0 since there is an index i {1,..., s} such that X i = ξ. Hence φ does not anish only on L ( M), and by definition it is a (s, 0) -form. (ii) We hae i ξ i ξ φ = 0 and taking into account (i), it results that i ξ φ (iii) If φ is a nonzero (s 1, 1) -form, then φ(x 1,..., X s ) 0 only if exactly one of the arguments is from the line distribution ξ. Then (i ξ φ)(x 1,..., X s 1 ) 0 for Ω s 1,0 some ertical ector fields X 1,..., X s 1 Γ(L ( M)).

7 A V -COHOMOLOGY WITH RESPECT TO COMPLEX LIOUVILLE DISTRIBUTION 157 (i) Let α be a form like in hypothesis and X 1,..., X s, s arbitrary ertical ector fields from Γ(L ( M)). Then φ(x 1,..., X s ) = (ω 0 α)(x 1,..., X s ) = σ S s ε(σ)ω 0 (X σ(1) )α(x σ(2),..., X σ(s) ) But, ω 0 anishes on L ( M) and for the all arguments X 1,..., X s Γ(L ( M)) the all terms of the aboe sum anish. Then by (3.4), we hae φ Ω s 1,1 Proposition 3.3. For any ertical (0, 0, 0, s) -form φ there are φ 1 Ω s,0 ( M) and φ 2 Ω s 1,1 ( M) such that φ = φ 1 + φ 2, uniquely. Proof. Let φ be a nonzero ertical (0, 0, 0, s) -form. If i ξ φ = 0, then by Proposition 3.2., we hae φ Ω s,0 So φ = φ + 0. If i ξ φ 0, then let φ 2 be the ertical (0, 0, 0, s) -form gien by ω 0 i ξ φ. By Proposition 3.2. (i), it results φ 2 is a (s 1, 1) -form. Moreoer, putting φ 1 = φ φ 2, we hae i ξ φ 1 = i ξ φ i ξ (ω 0 i ξ φ) = i ξ φ ω 0 (ξ)i ξ φ = 0 since ω 0 (ξ) = 1. So, φ 1 is a (s, 0) -form and φ 1, φ 2 are unique defined by φ. Obiously φ = φ 1 + φ 2. We hae to remark that only the zero form can be simultaneous a (s, 0) - and a (s 1, 1) -form, respectiely. The aboe Proposition leads to the decomposition (3.5) Ω s ( M) = Ω s,0 ( M) Ω s 1,1 ( M) A consequence of the Propositions 3.2. and 3.3. is Proposition 3.4. Let φ be a (0, 0, 0, s) -form. We hae the equialence (3.6) φ Ω s 1,1 Moreoer, the form α is uniquely determined. ( M) α Ω s 1,0 ( M), φ = ω 0 α Taking into account the characterization gien in Proposition 3.2. (i) and the relation (3.6), it follows Proposition 3.5. We hae the following facts (i) If φ Ω s,0 ( M) and ψ Ω t,0 ( M), then φ ψ Ω s+t,0 (ii) If φ Ω s,1 ( M) and ψ Ω t,0 ( M), then φ ψ Ω s+t,1 (iii) If φ Ω s,1 ( M) and ψ Ω t,1 ( M), then φ ψ = 0. Example 3.1. (i) ω 0 Ω 0,1 ( M) since there is the (0, 0) -form, the constant 1 function on M, such that ω 0 = ω 0 1. (ii) θ i = δη i η i ω 0 Ω 1,0 ( M), for each i = 1,..., n. Indeed θ i (ξ) = δη i (ξ) ω 0 (ξ)η i = 0 so i ξ θ i = 0. We hae to remark that the ertical (0, 0, 0, 1) -forms {θ i }, i = 1,..., n are linear dependent, since t i θ i = 0. (iii) i ξ (θ i θ j )(X) = θ i (ξ)θ j (X) θ j (ξ)θ i (X) = 0, for any ertical ector field X Γ(V ( M)), hence θ i θ j Ω 2,0

8 158 ADELINA MANEA AND CRISTIAN IDA Proposition 3.6. The conjugated ertical differential operator d has the following property: for any (s 1, 1) -form φ, d φ is a (s, 1) -form. Proof. Let φ be a (s 1, 1) -form. By (3.6), there is a (s 1, 0) -form α such that φ = ω 0 α. By Proposition 3.3. we hae also that α = i ξ φ. Taking into account that ω 0 is an d -exact form, it follows d φ = d (ω 0 α) = ω 0 d α = ω 0 β 1 ω 0 β 2 where β 1 and β 2 are the (s, 0)- and (s 1, 1) -forms, respectiely, components of the (0, 0, 0, s) -form d α. By (3.6) we hae β 2 = ω 0 γ with γ Ω s 1,0 ( M), so d φ = ω 0 β 1. Then d φ Ω s,1 We can write (3.7) d (Ω s 1,1 ( M)) Ω s,1 ( M) Now, we can consider p 1 and p 2 the projections of the module Ω s ( M) on its direct sumands from the relation (3.5), namely (3.8) p 1 : Ω s ( M) Ω s,0 ( M), p 1 φ = φ ω 0 i ξ φ (3.9) p 2 : Ω s ( M) Ω s 1,1 ( M), p 2 φ = ω 0 i ξ φ for any φ Ω s For an arbitrary (0, 0, 0, s) -form φ, we hae d φ = d (p 1 φ) + d (p 2 φ). The relation (3.5) shows that d (p 2 φ) is a (s, 1) -form, hence p 1 d (p 2 φ) = 0. It results (3.10) p 1 d φ = p 1 d (p 1 φ), p 2 d φ = p 2 d (p 1 φ) + p 2 d (p 2 φ) The aboe relations proe that (3.11) d (Ω s,0 ( M)) Ω s+1,0 ( M) Ω s,1 ( M) which allows to define the following operators: (3.12) d 1,0 : Ω s,0 ( M) Ω s+1,0 ( M), d 1,0φ = p 1 d φ (3.13) d 0,1 : Ω s,0 so that ( M) Ω s,1 ( M), d 0,1φ = p 2 d φ (3.14) d Ω s,0 ( M) = d 1,0 + d 0,1 Proposition 3.7. The operator d 1,0 satisfies (i) d 1,0(φ ψ) = d 1,0φ ψ + ( 1) s φ d 1,0ψ, φ Ω s,0 ( M), ψ Ω t,0 (ii) (d 1,0) 2 = 0. Proof. (i) Let φ Ω s,0 ( M) and ψ Ω t,0 According to [12], we hae and by (3.14) it follows d (φ ψ) = d φ ψ + ( 1) s φ d ψ d 1,0(φ ψ) + d 0,1(φ ψ) = d 1,0φ ψ + d 0,1φ ψ + ( 1) s φ d 1,0ψ + ( 1) s φ d 0,1ψ By equating the (s + t + 1, 0) components in the both members of aboe relation, we get the desired result.

9 A V -COHOMOLOGY WITH RESPECT TO COMPLEX LIOUVILLE DISTRIBUTION 159 (ii) Let φ be a (s, 0) -form. By (3.8) and (3.12) we hae that d 1,0φ = d φ ω 0 i ξ (d φ). Thus, using (d ) 2 = 0, d ω 0 = 0 and i ξ ω 0 = 1, by direct calculations, one gets (d 1,0) 2 φ = d 1,0(d φ) d 1,0(ω 0 i ξ (d φ)) = d (ω 0 i ξ (d φ)) + ω 0 i ξ (d (ω 0 i ξ (d φ))) = ω 0 d (i ξ (d φ)) + ω 0 i ξ ( ω 0 d (i ξ (d φ))) = ω 0 d (i ξ (d φ)) ω 0 d (i ξ (d φ)) = 0 Example 3.2. (i) For a (0, 0, 0, 1) -form φ, we hae p 1 φ = φ φ(ξ)ω 0 and p 2 φ = φ(ξ)ω 0. (ii) Let f F( M) and d f =. k (f)δη k its conjugated ertical deriatie. Locally, we hae d 0,1f = p 2 d f = (d f)(ξ)ω 0 = ξ(f)ω 0 and d 1,0f = p 1 d f = d f (d f)(ξ)ω 0 =. k (f)δη k η k. k (f)ω 0 =. k (f)θ k where θ k are the (1, 0) -forms gien in Example 3.1. Moreoer, taking into account the relation (2.4) and the fact t i θ i = 0, it results that locally (3.15) d 1,0f = (X i f)θ i We hae d 1,0η k = (X i η k )θ i = δ k i θi t i ξ(η k )θ i = θ k (t i θ i )η k = θ k so the (1, 0) -forms θ k are exactly the d 1,0 -deriaties of the local coordinates η k, for all k = 1,..., n. (iii) The (2, 0) -forms d 1,0η k d 1,0η j are d 1,0 -closed, for all j, k = 1,..., n. Let us consider an arbitrary (0, 0, 0, 1) -form on M. It is locally gien in U by φ = φ i δη i, with φ i F(U) such that in U U ϕ we hae φ = zi j z j φ i. By the Proposition 3.2., φ is a (1, 0) -form on M iff i ξ φ = 0 which is equialent locally with φ i η i = 0. Then, locally we hae φ = φ i δη i = φ i (d 1,0η i + η i ω 0 ) = φ i d 1,0η i + (φ i η i )ω 0 = φ i d 1,0η i Conersely, the expression locally gien by φ i d 1,0η i, with the functions φ i satisfying φ = zi j z j φ i is a (1, 0) -form because d 1,0η i (ξ) = 0, for all i = 1,..., n. 4. A d 1,0 -cohomology In [12] a classical theory of de Rham cohomology is deeloped for the conjugated ertical differential d. The sequence O Φ 0 i F 0 d F 1 d F 2 d... d...,

10 160 ADELINA MANEA AND CRISTIAN IDA is a fine resolution for the sheaf Φ 0 of germs of d -closed functions on M, where F s are the sheaes of germs of ertical (0,0,0,s)-forms. It is proed a de Rham type theorem for the -cohomology groups of the complex Finsler manifold: H s ( M, Φ 0 ) Z s ( M)/B s ( M) where Z s ( M) is the space of d -closed (0, 0, 0, s) -forms and B s ( M) is the space of d -exact (0, 0, 0, s) -forms globally defined on M. In this section we define new cohomology groups on M and we study the relations between these groups and H s ( M, Φ 0 ). Definition 4.1. We say that a (s, 0) -form φ is d 1,0 -closed if d 1,0φ = 0 and it is called d 1,0 -exact if φ = d 1,0ψ for some ψ Ω s 1,0 An important property of the operator d 1,0 is a Grothendieck-Dolbeault type lemma, namely Theorem 4.1. Let φ Ω s,0 (U) be a d 1,0 -closed form and s 1. Then there exists ψ Ω s 1,0 (U ), U U and such that φ = d 1,0ψ on U. Proof. Let φ Ω s,0 (U) such that 1,0φ = 0. Then d d φ = d 1,0φ + d 0,1φ = d 0,1φ = ω 0 i ξ (d φ) so d φ = 0 (modulo terms containing ω 0 ). Hence on the space ω 0 = 0 we hae that φ is d -closed. But the operator d satisfies a Grothendieck-Dolbeault type lemma (see [12]), so there exists a (0, 0, 0, s 1) -form τ defined on U U such that (4.1) φ = d τ + λ ω 0, λ Ω s 1 (U ) Following the Proposition 3.3. we hae that τ = τ 1 + ω 0 i ξ τ, with τ 1 = p 1 τ Ω s 1,0 (U ). Now, the relation (4.1) become Here φ Ω s,0 Ω s,0 (U ) Ω s 1,1 φ = d τ 1 ω 0 d i ξ τ + λ ω 0 (U ), ω 0 (λ + d i ξ τ) Ω s 1,1 (U ) and d τ 1 = d 1,0τ 1 + d 0,1τ 1 (U ). It results φ = d 1,0τ 1 on U. Let Φ be the sheaf of germs of functions on M which satisfies d 1,0f = 0 and F s,0 be the sheaf of germs of (s, 0) -forms on M. We denote by i : Φ F 0,0 be the natural inclusion. The sheaes F s,0 are fine and taking into account the Theorem 4.1., it follows that the sequence of sheaes 0 Φ i F 0,0 d 1,0 F 1,0 d 1,0... d 1,0 F s,0 d 1,0... is a fine resolution of Φ and we denote by H s ( M, Φ ) the cohomology groups of M with the coefficients in the sheaf Φ. Then, we obtain a de Rham type theorem, namely Theorem 4.2. The -cohomology groups with respect to the operator d 1,0 of (s, 0) -forms on a complex Finsler manifold are gien by (4.2) H s ( M, Φ ) Z s,0 s,0 ( M)/B ( M)

11 A V -COHOMOLOGY WITH RESPECT TO COMPLEX LIOUVILLE DISTRIBUTION 161 where Z s,0 ( M) is the space of d 1,0 -closed (s, 0) -forms and B s,0 ( M) is the space of d 1,0 -exact (s, 0) -forms globally defined on M. By (3.7), the de Rham complex O F 0,0 d ( M) Ω 1 ( M) d Ω 2 ( M) d... d Ω s ( M) d..., admits the subcomplex O Φ ( M) d Ω 0,1 d ( M) Ω 1,1 d ( M)... d Ω s,1 d ( M)..., We denote by Z s,1 ( M) and B s,1 ( M) the spaces of the d -closed and d -exacts (s, 1)-forms, respectiely, and let H s,1 ( M) = Z s,1 s,1 (4.3) ( M)/B be the s-cohomology group of the last complex. Theorem 4.3. The cohomology groups H s,1 ( M) and H s ( M, Φ ) are isomorphic. Proof. By Proposition 3.4 we can define the map ζ : Z s,1 ( M) Z s,0 ( M), ζ(φ) = α for α Ω s,0 ( M) such that φ = α ω 0. It is a well-defined map since the equality 0 = d φ = d α ω 0 = d 1,0α ω 0 + d 0,1α ω 0 = d 1,0α ω 0 implies d 1,0α = 0. Moreoer, ζ is a bijectie morphism of groups and ζ(b s,1 ( M)) = B s,0 Indeed, for φ B s,1 ( M), there is θ Ω s 1,1 ( M) such that φ = d θ. By (3.6), there are α Ω s,0 ( M), β Ω s 1,0 ( M) such that φ = α ω 0 and θ = β ω 0. Then, we hae α ω 0 = d (β ω 0 ) = d 1,0β ω 0 It follows α B s,0 Conersely, α = d 1,0β implies α ω 0 = d (β ω 0 ). We obtain that ζ : H s,1 ( M) Z s,0 s,0 ( M)/B ( M), ζ ([φ]) = [ζ(φ)], for φ Z s,1 ( M), is bijectie. Finally, we remark that the projection p 1 from (3.8) induces the morphism p 1 : Z( M) s Z s,0 ( M) Indeed, taking into account the Propositions 3.3. and 3.6. and the relations (3.7) and (3.14), we obtain for φ Z s( M): 0 = d φ = d (φ 1 + φ 2 ) = d 1,0φ 1 + d 0,1φ 1 + d φ 2 which implies d 1,0φ 1 = 0 since d 1,0φ 1 Ω s+1,0 ( M) and d 0,1φ 1 + d φ 2 Ω s,1 Moreoer, for eery d -exact form φ = d θ, with θ Ω s 1 ( M), we hae by Proposition 3.3. that φ = φ 1 + φ 2 = d (θ 1 + θ 2 ) = d 1,0θ 1 + d 0,1θ 1 + d θ 2 Equating the (s, 0)-components we obtain φ 1 = d 1,0θ 1. So, the morphism p 1 satifies p 1(B s( M)) = B s,0 Hence p 1 induces a morphism of cohomology groups p 1 : Z s ( M)/B s ( M) Z s,0 s,0 ( M)/B ( M)

12 162 ADELINA MANEA AND CRISTIAN IDA which is not always injectie. References [1] Abate, M., Patrizio, G., Finsler metrics-a global approach, Lectures Notes in Math., 1591, Springer-Verlag, [2] Aikou, T., The Chern-Finsler connection and Finsler-Kähler manifolds, Ad. Stud. in Pure Math. 48, (2007), [3] Bejancu, A., Farran, H. R., On The Vertical Bundle of a pseudo-finsler Manifold, Int. J. Math. and Math. Sci. 22 no. 3, (1997), [4] Bejancu, A., Farran, H. R., Finsler Geometry and Natural Foliations on the Tangent Bundle, Rep. on Mathematical Physics 58 no. 1 (2006), [5] Kobayashi, S., Negatie Vector Bundles and Complex Finsler Sructures, Nagoya Math. J. 57 (1975), [6] Kobayashi, S., Differential Geometry of Complex Vector Bundles, Iwanami Princeton Uni. Press, [7] Kobayashi, S., Complex Finsler Vector Bundles, Contemporary Math., 196 (1996), [8] Manea, A., Some new types of ertical 2 jets on the tangent bundle of a Finsler manifold, U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 1 (2010), [9] Manea, A., A de Rham theorem for a Liouille foliation on T M 0 oer a Finsler manifold M, Differential Geometry - Dynamical Systems, 13, (2011), [10] Miron, R., Anastasiei, M., The Geometry of Lagrange Spaces. Theory and Applications, Kluwer Acad. Publ. 59, [11] Munteanu, G., Complex spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Acad. Publ., 141 FTPH, [12] Pitiş, G., Munteanu, G., V-cohomology of complex Finsler manifolds, Studia Uni., Babeş- Bolyai, XLIII (3) (1998), [13] Vaisman, I., Variétés riemanniene feuilletées, Czechosloak Math. J., 21 (1971), [14] Vaisman, I., Sur la cohomologie des arietés analytiques complexes feuilletées, C. R. Acad. Sc. Paris, t. 273 (1971), [15] Vaisman, I., Cohomology and differential forms, New York, M. Dekker Publ. House, Department of Mathematics and Informatics, Transilania Uniersity of Braşo, ROMÂNIA address: amanea28@yahoo.com; cristian.ida@unitb.ro