Cohomology of holomorphic line bundles and Hodge symmetry on Oeljeklaus-Toma manifolds
aa r X i v : . [ m a t h . AG ] F e b COHOMOLOGY OF HOLOMORPHIC LINE BUNDLES AND HODGESYMMETRY ON OELJEKLAUS-TOMA MANIFOLDS
HISASHI KASUYA
Abstract.
We prove the Hodge symmetry type result on the Dolbeault cohomology ofOeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Con-sequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles. Introduction
For positive integers s, t , let K be a finite extension field of Q of degree s + 2 t admittingembeddings σ , . . . σ s , σ s +1 , . . . , σ s +2 t into C such that σ , . . . , σ s are real embeddings and σ s +1 , . . . , σ s +2 t are complex ones satisfying σ s + i = ¯ σ s + i + t for 1 ≤ i ≤ t . Denote by O K the ring of algebraic integers of K . In [9], for a free subgroup U of rank s in the group ofunits in O K satisfying certain conditions related to the embeddings σ , . . . σ s , σ s +1 , . . . , σ s +2 t ,Oeljeklaus and Toma constructs a ( s + t )-dimensional complex manifold X ( K, U ) whosefundamental group is the semi-direct product U ⋉ O K of finitely generated free abeliangroups U and O K . We call this complex manifold an Oeljeklaus-Toma (OT) manifold. Forany OT-manifold X ( K, U ), the Hodge symmetrydim H p,q ( X ( K, U )) = dim H q,p ( X ( K, U ))on the Dolbeault cohomology does not hold (see [9, Proposition 2.5]). Hence every OT-manifold X ( K, U ) is a non-K¨ahler complex manifold. Meanwhile, in [10], Otiman and Tomashow that the Hodge decompositiondim H r ( X ( K, U ) , C ) = X p + q = r dim H p,q ( X ( K, U ))holds for any OT-manifold X ( K, U ).Consider the set Hom( U, C ∗ ) of group homomorphisms from U to C ∗ . Correspondingrepresentations of the fundamental group of X ( K, U ) to flat bundles over X ( K, U ), this setis identified with a set A ( U ) of isomorphism classes of flat complex line bundles over X ( K, U ).For E ∈ A ( U ), we consider the de Rham cohomology H ∗ ( X ( K, U ) , E ) with values in E . Wehave the following result. Proposition 1.1.
For any integer r , we have dim r ^ C s +2 t = X E ∈A ( U ) dim H r ( X ( K, U ) , E ) . Mathematics Subject Classification.
Key words and phrases.
Oeljeklaus-Toma manifold, solvmanifold, Dolbeault cohomology, holomorphicLine bundle.
As explained in [5], every OT-manifold is diffeomorphic to a solvmanifold. This propositionis a consequence of the cohomology computation of solvmanifolds in [7].Regarding each E ∈ A ( U ) as a holomorphic line bundle over X ( K, U ), we consider theDolbeault cohomology H p,q ( X ( K, U ) , E ) = H q ( X ( K, U ) , Ω p ( E )). In this paper, we provethe following Hodge symmetry type result. Theorem 1.2.
For any integers p, q , we have dim p ^ C s + t ⊗ q ^ C s + t = X E ∈A ( U ) dim H p,q ( X ( K, U ) , E ) . More precisely, we can give explicit harmonic representatives of L A ( U ) H p,q ( X ( K, U ) , E ).In particular, we know the vanishing or non-vanishing of the cohomology H p,q ( X ( K, U ) , E )for any E ∈ A ( U ). For a positive integer s , we denote [ s ] = { , , . . . , s } and we call a subsetin [ s ] with the natural order a multi-index. Theorem 1.3.
Let E be a flat complex line bundle over an OT-manifold X ( K, U ) corre-sponding to ρ ∈ Hom( U, C ∗ ) . Then H p,q ( X ( K, U ) , E ) = 0 if and only if for some multi-indices I ⊂ [ s ] , K, L ⊂ [ t ] with | I | + | K | = p and | L | ≤ q , wehave ρ ( u ) = Y i ∈ I σ i ( u ) Y j ∈ J σ s + j ( u ) Y k ∈ K σ s + t + k ( u ) for any u ∈ U . If ρ ( u ) = Y i ∈ I σ i ( u ) Y j ∈ J σ s + j ( u ) Y k ∈ K σ s + t + k ( u ) for any u ∈ U , then we have dim H p,q ( X ( K, U ) , E ) ≥ (cid:18) sq − | L | (cid:19) where (cid:18) nk (cid:19) means the number of k -combinations. Obviously we have the following consequence.
Corollary 1.4.
Let E be a flat complex line bundle over an OT-manifold X ( K, U ) corre-sponding to ρ ∈ Hom( U, C ∗ ) . Then, E admits a non-zero holomorphic section if and only if ρ is trivial. As a corollary of Theorem 1.3, we also obtain [1, Theorem 3.1] (see Remark 5.3).
Remark 1.5.
Our main results are obtained by using the solvmanifold presentations of OTmanifolds. We give statements in terms of solvmanifolds (Proposition 3.1, Theorem 4.1 andTheorem 5.1) implying Proposition 1.1, Theorem 1.2 and Theorem 1.3.We notice that Theorem 1.2 is deeper than Proposition 1.1. In [5] , we also compute thecohomology of complex parallelizable solvmanifolds with values in holomorphic vector bundles.OT-manifolds do not admit non-zero holomorphic vector field ( [9, Proposition 2.5] ) and theyare far from complex parallelizable solvmanifolds. To prove Theorem 1.2, we use the resultin [10] given by the analysis on Cousin groups.
INE BUNDLES AND HODGE SYMMETRY ON OELJEKLAUS-TOMA MANIFOLDS 3 Cohomology of solvmanifolds
Let G be a simply connected real solvable Lie group with a lattice Γ and g the Lie algebraof G . Denote by A ∗ (Γ \ G ) the de Rham complex of the solvmanifold Γ \ G . We identify thede Rham complex A ∗ (Γ \ G ) of Γ \ G with the subcomplex of the de Rham complex A ∗ ( G ) of G consisting of the left-Γ-invariant differential forms. Consider the cochain complex V g ∗ ofthe Lie algebra g . We identify V g ∗ with the subcomplex of the de Rham complex A ∗ ( G ) of G consisting of the left- G -invariant differential forms. Hence, we consider V g ∗ as a subcomplexof A ∗ (Γ \ G ).Let N be the nilradical (i.e. maximal connected nilpotent normal subgroup) of G . Denote A ( G,N ) = { α ∈ Hom( G, C ∗ ) | α | N = 1 } . For α ∈ A ( G,N ) , we consider the Γ-action on G × C so that for γ ∈ Γ and ( g, c ) ∈ G × C , γ · ( g, c ) = ( γg, α ( γ ) c ). Define the flat complex linebundle E α = Γ \ ( G × C ) over the solvmanifold Γ \ G . We have the global section v α inducedby the section ( g, α ( g )) of G × C . Denote by A ∗ (Γ \ G, E α ) the de Rham complex with valuesin the line bundle E α . Since v α trivializes E α , we have A ∗ (Γ \ G, E α ) = A ∗ (Γ \ G ) ⊗ h v α i and dv α = α − dαv α . We have the subcomplex V g ∗ C ⊗ h v α i ⊂ A ∗ (Γ \ G, E α ) = A ∗ (Γ \ G ) ⊗ h v α i .The cochain complex V g ∗ C ⊗ h v α i is the cochain complex of the Lie algebra g associated withthe representation α ∈ Hom( G, C ∗ ).We define A ( G,N ) (Γ) by the set { E α } of all the isomorphism classes of flat line bundles E α associated with α ∈ A ( G,N ) . This set is identified with the set { α | Γ ∈ Hom(Γ , C ∗ ) | α ∈A ( G,N ) } . Consider the direct sum M E α ∈A ( G,N ) (Γ) A ∗ (Γ \ G, E α ) . By the natural isomorphisms E α ⊗ E β ∼ = E αβ for α, β ∈ A ( G,N ) , this direct sum is a differentialgraded algebra. By the above argument, we have the inclusion M α ∈A ( G,N ) ^ g ∗ ⊗ h v α i ⊂ M E α ∈A ( G,N ) (Γ) A ∗ (Γ \ G, E α ) . We have a simply connected nilpotent subgroup C ⊂ G such that G = C · N (see [2,Proposition 3.3]). Since C is nilpotent, the mapΦ : C ∋ c (Ad c ) s ⊗ α ( c ) ∈ Aut (cid:16)^ g ∗ C ⊗ h v α i (cid:17) is a homomorphism where (Ad c ) s is the semi-simple part of the Jordan decomposition of theadjoint operator. We denote by (cid:16)^ g ∗ C ⊗ h v α i (cid:17) Φ( C ) the subcomplex of V g ∗ C ⊗ h v α i consisting of the Φ( C )-invariant elements. [7, Theorem 1.4,Lemma 5.2] says that the inclusion M α ∈A ( G,N ) (cid:16)^ g ∗ C ⊗ h v α i (cid:17) Φ( C ) ⊂ M E α ∈A ( G,N ) (Γ) A ∗ (Γ \ G, E α )induces a cohomology isomorphism.We have a basis X , . . . , X n of g C such that (Ad c ) s = diag( α ( c ) , . . . , α n ( c )) for all c ∈ C .Let x , . . . , x n be the basis of g ∗ C which is dual to X , . . . , X n . By G = C · N , we have H. KASUYA
G/N = C/C ∩ N and hence we have A ( G,N ) = A C,C ∩ N = { α ∈ Hom( C, C ∗ ) | α | C ∩ N = 1 } . Fora multi-index I = { i , . . . , i p } ⊂ [ n ] we write x I = x i ∧ · · · ∧ x i p , and α I = α i · · · α i p . Weconsider the basis { x I ⊗ v α } I ⊂ [ n ] of V g ∗ C ⊗ h v α i . Since the actionΦ : C → Aut (cid:16)^ g ∗ C ⊗ h v α i (cid:17) is the semi-simple part of (Ad ⊗ α ) | C , we haveΦ( a )( x I ⊗ v α ) = α − I αx I ⊗ v α . Hence we have M α ∈A ( G,N ) (cid:16)^ g ∗ C ⊗ h v α i (cid:17) Φ( C ) = h x I ⊗ v α I i I ⊂ [ n ] = ^ h x ⊗ v α , . . . , x n ⊗ v α n i . It is known that the differential graded algebra ^ h x ⊗ v α , . . . , x n ⊗ v α n i is identified with the cochain complex of certain nilpotent Lie algebra determined by thesolvable Lie algebra g (see [7, Remark 4] and [6]).3. de Rham and Dolbeault cohomology of certain solvmanifolds Let s, t be positive integers. We consider the semi-direct product G = R s ⋉ φ ( R s ⊕ C t ) ofreal abelian Lie groups R s and R s ⊕ C t given by the homomorphism φ : R s → Aut( R s ⊕ C t )so that φ ( x )( y, z ) = ( e x y , . . . , e x s y s , e ψ ( x ) z , . . . , e ψ t ( x ) z t )for x = ( x , . . . , x s ) ∈ R s , ( y, z ) = ( y , . . . , y s , z , . . . , z t ) ∈ R s ⊕ C t and some non-zero linearfunctions ψ , . . . , ψ t : R s → C . G is a simply connected solvable Lie group. Suppose we havelattices Λ ⊂ R s and ∆ ⊂ R s ⊕ C t so that for every λ ∈ Λ the automorphism φ ( λ ) on R s ⊕ C t preserves ∆. Then the subgroup Γ = Λ ⋉ φ ∆ ⊂ G is a cocompact discrete subgroup of G .Consider the solvmanifold Γ \ G .We have a basis dx , . . . , dx s , e − x dy , . . . , e − x s dy s , e − ψ ( x ) dz , . . . , e − ψ t ( x ) dz t , e − ¯ ψ ( x ) d ¯ z , . . . , e − ¯ ψ t ( x ) d ¯ z t of g ∗ C = g ∗ ⊗ C .On G = R n ⋉ φ ( R s ⊕ C t ), the nilradical N is R s ⊕ C t and we take the subgroup C = R s ⊂ G so that C · N = G . We apply the last section to the solvmanifold Γ \ G . By the last section,defining V = * dx , . . . , dx s ,e − x dy ⊗ v e x , . . . , e − x s dy s ⊗ v e xs ,e − ψ ( x ) dz ⊗ v e ψ x ) , . . . , e − ψ t ( x ) dz m ⊗ v e ψt ( x ) ,e − ¯ ψ ( x ) d ¯ z ⊗ v e ¯ ψ x ) , . . . , e − ¯ ψ t ( x ) d ¯ z t ⊗ v e ¯ ψt ( x ) + INE BUNDLES AND HODGE SYMMETRY ON OELJEKLAUS-TOMA MANIFOLDS 5 where we regard dx , . . . , dx s as 1-forms with values in the trivial line bundle, we have theinclusion ^ V ⊂ M E α ∈A ( G,N ) (Γ) A ∗ (Γ \ G, E α )which induces a cohomology isomorphism. We notice that we can identify A ( G,N ) withthe set Hom( R s , C ∗ ) of Lie group homomorphisms from R s to C ∗ and the set A ( G,N ) (Γ)is equal to the set of isomorphism classes of flat complex line bundles over Γ \ G given byhomomorphisms in Hom(Λ , C ∗ ). We can easily check that the differential on V is 0. Hencewe have the following: Proposition 3.1.
We have an isomorphism ^ V ∼ = M E α ∈A ( G,N ) H ∗ (Γ \ G, E α ) . Regarding 1-forms α = dx + √− e − x dy , . . . , α s = dx s + √− e − x s dy s , β = e − ψ ( x ) dz , . . . , β t = e − ψ t ( x ) dz t as (1 , \ G , we have a left- G -invariant almost complex structure J on Γ \ G . Wecan easily check that J is integrable. We consider the Dolbeault complex ( A ∗ , ∗ (Γ \ G ) , ¯ ∂ ) ofthe complex manifold (Γ \ G, J ). We have¯ ∂α i = −
12 ¯ α i ∧ α i , ¯ ∂ ¯ α i = 0 , ¯ ∂β i = − ψ i ( ¯ α ) ∧ β i and ¯ ∂ ¯ β i = −
12 ¯ ψ i ( ¯ α ) ∧ ¯ β i where ψ i ( ¯ α ) and ¯ ψ i ( ¯ α ) are (0 , ψ i ( x ) and ¯ ψ i ( x ) byputting x = ¯ α = ( ¯ α , . . . , ¯ α s ). Remark 3.2.
We consider the holomorphic tangent bundle Θ and holomorphic cotangentbundle Ω of (Γ \ G, J ) . Then, α , . . . , α s , β , . . . , β t is a global C ∞ -frame of Ω . Hence wehave an isomorphism Ω ∼ = E e − x ⊕ · · · ⊕ E e − xs ⊕ E e − ψ x ) ⊕ · · · ⊕ E e − ψt ( x ) of holomorphic vector bundles. By this, we also have Θ ∼ = E e x ⊕ · · · ⊕ E e xs ⊕ E e ψ x ) ⊕ · · · ⊕ E e ψt ( x ) . For any e Ψ( x ) ∈ A ( G,N ) = Hom( R s , C ∗ ) associated with a complex valued linear functionΨ( x ) on R s , we regard the flat line bundle E e Ψ( x ) as a holomorphic line bundle over thecomplex manifold (Γ \ G, J ). We have ¯ ∂v e Ψ( x ) = Ψ( ¯ α ) ⊗ v e Ψ( x ) . Define W = (cid:28) α ⊗ v e x , . . . , α s ⊗ v e xs ,β ⊗ v e ψ x ) , . . . , β t ⊗ v e ψt ( x ) (cid:29) , W = (cid:28) ¯ α , . . . , ¯ α s , ¯ β ⊗ v e ¯ ψ x ) , . . . , ¯ β t ⊗ v e ¯ ψt ( x ) (cid:29) (3.1)where we regard ¯ α , . . . , ¯ α s as 1-forms with values in the trivial line bundle. We consider thesubspace p ^ W ⊗ q ^ W ⊂ M E α ∈A ( G,N ) (Γ) A p,q (Γ \ G, E α ) . Define the left- G -invariant Hermitian metric h G = α · ¯ α + · · · + α s · ¯ α s + β · ¯ β + · · · + β t · ¯ β t . H. KASUYA
Define the hermitian metric h α on each E α ∈ so that h α ( v α , v α ) = 1. We notice thatfor α, α ′ ∈ A ( G,N ) , if E α = E α ′ , then h α = h α ′ since E α = E α ′ if and only if α | Γ = α ′| Γ and hence α − α ′ is unitary. We consider the Hodge star operator ¯ ∗ : A p,q (Γ \ G, E α ) → A s + t − p,s + t − q (Γ \ G, E ∗ α ) associated with this metric. Then we have¯ ∗ (cid:0) α I ∧ ¯ α J ∧ β K ∧ ¯ β L ⊗ v e Ψ IK ¯ L ( x ) (cid:1) = ± α ˇ I ∧ ¯ α ˇ J ∧ β ˇ K ∧ ¯ β ˇ L ⊗ v e − Ψ IK ¯ L ( x ) where for a multi-indices I, J ⊂ [ s ] , K, L ⊂ [ t ] we writeΨ IK ¯ L ( x ) = X j ∈ I x j + X k ∈ K ψ k ( x ) + X l ∈ L ¯ ψ l ( x ) , ˇ I = [ s ] − I , ˇ J = [ s ] − J , ˇ K = [ t ] − K and ˇ L = [ t ] − L . Since G admits a lattice Γ, G isunimodular (see [12, Remark 1.9]). This impliesexp(Ψ [ s ][ t ][ t ] ( x )) = exp X i ∈ [ s ] x i + X k ∈ [ t ] ψ k ( x ) + X l ∈ [ t ] ¯ ψ l ( x ) = 1 . Thus exp(Ψ ˇ I ˇ K ˇ L ( x )) = exp( − Ψ IK ¯ L ( x )) . By this, we can say that the Hodge star operator ¯ ∗ preserves the space V W ⊗ V W (compare[8, Lemma 2.3]).We can easily check that the Dolbeault operator on W and W is 0. Hence, V W ⊗ V W consists of harmonic forms associated with the Dolbeault operator. This implies the followingresult. Proposition 3.3.
We have an injection p ^ W ⊗ q ^ W ֒ → M E α ∈A ( G,N ) (Γ) H p,q (Γ \ G, E α ) hence we have dim p ^ C s + t ⊗ q ^ C s + t ≤ X E α ∈A ( G,N ) (Γ) dim H p,q (Γ \ G, E α ) . Oeljeklaus-Toma manifolds
For positive integers s, t , let K be a finite extension field of Q of degree s + 2 t admittingembeddings σ , . . . σ s , σ s +1 , . . . , σ s +2 t into C such that σ , . . . , σ s are real embeddings and σ s +1 , . . . , σ s +2 t are complex ones satisfying σ s + i = ¯ σ s + i + t for 1 ≤ i ≤ t . Let O K be the ringof algebraic integers of K , O ∗ K the group of units in O K and O ∗ + K = { a ∈ O ∗ K : σ i ( a ) > ≤ i ≤ s } . Define σ : O K → R s × C t by σ ( a ) = ( σ ( a ) , . . . , σ s ( a ) , σ s +1 ( a ) , . . . , σ s + t ( a ))for a ∈ O K . Define l : O ∗ + K → R s + t by l ( a ) = (log | σ ( a ) | , . . . , log | σ s ( a ) | , | σ s +1 ( a ) | , . . . , | σ s + t ( a ) | ) INE BUNDLES AND HODGE SYMMETRY ON OELJEKLAUS-TOMA MANIFOLDS 7 for a ∈ O ∗ + K . Then by Dirichlet’s units theorem, l ( O ∗ + K ) is a lattice in the vector space L = { x ∈ R s + t | P s + ti =1 x i = 0 } . Consider the projection p : L → R s given by the first s coordinate functions. Then we have a subgroup U with the rank s of O ∗ + K such that p ( l ( U ))is a lattice in R s . Write l ( U ) = Z v ⊕ · · · ⊕ Z v s for generators v , . . . v s of l ( U ). For thestandard basis e , . . . , e s + t of R s + t , we have a regular real s × s -matrix ( a ij ) and s × t realconstants b jk such that v i = s X j =1 a ij ( e j + √− t X k =1 b jk e s + k )for any 1 ≤ i ≤ s . Consider the complex upper half plane H = { z ∈ C : Im z > } = R × R > .We have the left action of U ⋉ O K on H s × C t such that( a, b ) · ( x + √− y , . . . , x s + √− y s , z , . . . , z t )= ( σ ( a ) x + σ ( b ) + √− σ ( a ) y , . . . , σ s ( a ) x s + σ s ( b ) + √− σ s ( a ) y s ,σ s +1 ( a ) z + σ s +1 ( b ) , . . . , σ s + t ( a ) z t + σ s + t ( b )) . In [9] it is proved that the quotient X ( U, K ) = U ⋉ O K \ H s × C t is compact. Actuallywe have the real fiber bundle X ( U, K ) → U \ ( R > ) s with the fiber σ ( O K ) \ ( R s × C t ) andboth the base U \ ( R > ) s and the fiber σ ( O K ) \ ( R s × C t ) are real tori. We call this complexmanifold an Oeljeklaus-Toma (OT) manifold.As in [5], we present OT-manifolds as solvmanifolds considered in the last section. For a ∈ U and ( x , . . . , x s ) = p ( l ( a )) ∈ p ( l ( U )), since l ( U ) is generated by the basis v , . . . , v s asabove, l ( a ) is a linear combination of e + P tk =1 b k e s + k , . . . , e s + P tk =1 b sk e s + k and hence wehave l ( a ) = s X i =1 x i ( e i + t X k =1 b ik e s + k ) = ( x , . . . , x s , s X i =1 b i x i , . . . , s X i =1 b it x i ) . By 2 log | σ s + k ( a ) | = P si =1 b ik x i , we can write σ s + k ( a ) = e P si =1 b ik x i + √− P si =1 c ik x i for some c ik ∈ R . We consider the Lie group G = R s ⋉ φ ( R s × C t ) with φ ( x , . . . , x s ) = diag( e x , . . . , e x s , e ψ ( x ) , . . . , e ψ t ( x ) )where ψ k = P si =1 b ik x i + P si =1 c ik x i . Then for ( x , . . . , x s ) ∈ p ( l ( U )), we have φ ( x , . . . , x s )( σ ( O K )) ⊂ σ ( O K ) . Write p ( l ( U )) = Λ and σ ( O K ) = ∆. Then, via the diffeomorphism H s × C t ∋ ( y + √− w , . . . , y s + √− w s , z , . . . , z t ) (log( w ) , . . . , log( w s ) , y , . . . y s , z , . . . , z t ) ∈ R s ⋉ φ ( R s × C t ) , the OT-manifold X ( U, K ) = U ⋉ O K \ H s × C t is identified with a complex solvmanifold(Γ \ G, J ) of the form as in the last section.
Theorem 4.1.
Define W and W as (3.1). An isomorphism p ^ W ⊗ q ^ W ∼ = M E α ∈A ( G,N ) (Γ) H p,q (Γ \ G, E α ) H. KASUYA holds. Hence we have dim p ^ C s + t ⊗ q ^ C s + t = X E α ∈A ( G,N ) (Γ) dim H p,q (Γ \ G, E α ) . Proof.
We regard H p,q (Γ \ G, E α ) as the sheaf cohomology H q (Γ \ G, Ω p ( E α )). We consider thereal fiber bundle π : Γ \ G → Λ \ R s over the real torus Λ \ R s with the real torus fiber ∆ \ ( R s × C t ). This fiber bundle is identified with the fiber bundle X ( U, K ) → U \ ( R > ) s with the fiber σ ( O K ) \ ( R s × C t ). Consider the Leray spectral sequence E ∗ , ∗∗ (Ω p ( E α )) associated with themap π : Γ \ G → Λ \ R s and the sheaf Ω p ( E α ). Then E a,b (Ω p ( E α )) = H a (Λ \ R s , R b π ∗ Ω p ( E α ))and E a,br (Ω p ( E α )) converges to H a + b (Γ \ G, Ω p ( E α )). The sheaf R b π ∗ Ω p ( E α ) over Λ \ R s isthe sheafification of the pre-sheaf such that each open set O ⊂ Λ \ R s corresponds to thevector space H b ( π − ( O ) , Ω p ( E α )). Since the flat bundle E α corresponds to a homomorphism α | Λ ∈ Hom (Λ , C ∗ ), as a sheaf on Γ \ G , E α is constant on π − ( O ) for sufficiently small openset O ⊂ Λ \ R s . Thus we have R b π ∗ Ω p ( E α ) ∼ = ( R b π ∗ Ω p ) ⊗ ˜ E α | Λ where ˜ E α | Λ is the local systemon Λ \ R s corresponding to α | Λ ∈ Hom (Λ , C ∗ ). [10, Lemma 4.3] says that R b π ∗ Ω p is a localsystem on the torus Λ \ R s so that locally R b π ∗ Ω p is isomorphic to V p C s + t ⊗ V b C t and R b π ∗ Ω p corresponds to a diagonal representation of Λ = p ( l ( U )) ∼ = U . Let E be a local system onΛ \ R s corresponding to a 1-dimensional complex representation of Λ. It is well known that H ∗ (Λ \ R s , E ) ∼ = V C s for trivial E otherwise H ∗ (Λ \ R s , E ) = 0. By this for any E , we have M β ∈ Hom (Λ , C ∗ ) H ∗ (Λ \ R s , E ⊗ ˜ E β ) = H ∗ (Λ \ R s , E ⊗ E − ) ⊕ M ˜ E β = E − H ∗ (Λ \ R s , E ⊗ ˜ E β ) ∼ = ^ C s . Thus, identifying A ( G,N ) (Γ) with Hom (Λ , C ∗ ), we have M E α ∈A ( G,N ) (Γ) E a,b (Ω p ( E α )) ∼ = M β ∈ Hom (Λ , C ∗ ) H a (Λ \ R s , ( R b π ∗ Ω p ) ⊗ ˜ E β ) ∼ = a ^ C s ⊗ p ^ C s + t ⊗ b ^ C t . We have M a + b = q M E α ∈A ( G,N ) (Γ) E a,b (Ω p ( E α )) ∼ = p ^ C s + t ⊗ q ^ C s + t . By Proposition 3.3, we have X a + b = q X E α ∈A ( G,N ) (Γ) dim E a,b (Ω p ( E α )) = dim p ^ C s + t ⊗ q ^ C s + t ≤ X E α ∈A ( G,N ) (Γ) dim H p,q (Γ \ G, E α ) . Thus, the Leray spectral sequence E ∗ , ∗∗ (Ω p ( E α )) degenerates at E -term and so we havedim p ^ C s + t ⊗ q ^ C s + t = X E α ∈A ( G,N ) (Γ) dim H p,q (Γ \ G, E α ) . Hence the injection in Proposition 3.3 is an isomorphism. (cid:3)
Since U ∼ = p ( l ( U )) = Λ is embedded in R s as a lattice, the set A ( G,N ) (Γ) is equal to theset A ( U ) of isomorphism classes of flat complex line bundles over X ( K, U ) = Γ \ G given byhomomorphisms in Hom( U, C ∗ ). We have the following consequence of Proposition 3.3. INE BUNDLES AND HODGE SYMMETRY ON OELJEKLAUS-TOMA MANIFOLDS 9
Corollary 4.2.
For any integer r , we have dim r ^ C s +2 t = X E ∈A ( U ) dim H r ( X ( K, U ) , E ) . Remark 4.3.
See [4] for the de Rham cohomology of OT-manifolds with values in trivialand some specific flat line bundles.
Theorem 4.1 gives the following statement.
Corollary 4.4.
For any integers p, q , we have dim p ^ C s + t ⊗ q ^ C s + t = X E ∈A ( U ) dim H p,q ( X ( K, U ) , E ) . Cohomology of holomorphic line bundles over OT-manifolds:vanishingand non-vanishing
For each E α ∈ A ( G,N ) (Γ), by Theorem 4.1, we have H p,q (Γ \ G, E α ) ∼ = (cid:28) α I ∧ ¯ α J ∧ β K ∧ ¯ β L ⊗ v e Ψ IK ¯ L ( x ) (cid:12)(cid:12)(cid:12)(cid:12) | I | + | K | = p and | J | + | L | = qE α = E e Ψ IK ¯ L ( x ) (cid:29) . Corollary 5.1. H p,q (Γ \ G, E α ) = 0 if and only if for some multi-indices I ⊂ [ s ] , K, L ⊂ [ t ] with | I | + | K | = p and | L | ≤ q , E α = E e Ψ IK ¯ L ( x ) . If E α = E e Ψ IK ¯ L ( x ) , then we have dim H p,q (Γ \ G, E α ) ≥ (cid:18) sq − | L | (cid:19) where (cid:18) nk (cid:19) means the number of k -combinations. We notice that E α = E e Ψ IK ¯ L ( x ) if and only if α ( x ) = e Ψ IK ¯ L ( x ) for any x ∈ Λ. For the trivial E α , this Corollary gives [8, Corollary 3.5].For u ∈ U with x = p ( l ( u )) ∈ p ( l ( U )) we have σ i ( u ) = e x i for 1 ≤ i ≤ s , σ s + k ( u ) = e ψ k ( x ) and σ s + t + k ( u ) = e ¯ ψ k ( x ) for 1 ≤ k ≤ s . Hence we have e Ψ IK ¯ L ( x ) = Y i ∈ I σ i ( u ) Y j ∈ J σ s + j ( u ) Y k ∈ K σ s + t + k ( u ) . Corollary 5.2.
Let E be a flat complex line bundle over an OT-manifold X ( K, U ) corre-sponding to ρ ∈ Hom( U, C ∗ ) . Then H p,q ( X ( K, U ) , E ) = 0 if and only if for some multi-indices I ⊂ [ s ] , K, L ⊂ [ t ] with | I | + | K | = p and | L | ≤ q , wehave ρ ( u ) = Y i ∈ I σ i ( u ) Y j ∈ J σ s + j ( u ) Y k ∈ K σ s + t + k ( u ) for any u ∈ U . If ρ ( u ) = Y i ∈ I σ i ( u ) Y j ∈ J σ s + j ( u ) Y k ∈ K σ s + t + k ( u ) for any u ∈ U , then we have dim H p,q ( X ( K, U ) , E ) ≥ (cid:18) sq − | L | (cid:19) . Remark 5.3.
This statement implies [1, Theorem 3.1] . Actually, for p = 0 and q = 1 , H , ( X ( K, U ) , E ) = 0 if and only if ρ is trivial or ρ = σ s + t +1 , . . . , σ s +2 t . This seems differentfrom [1, Theorem 3.1] . But we may remark that we use the left action but on the other handin [1] the right action is used. The correspondence between the right-quotient and left-quotientis given by the inverse. Example 5.4.
We consider the case t = 1. In this case, any U is a finite-index subgroup of O ∗ + K . For our solvmanifold presentation Γ \ G of an OT-manifold X ( K, U ), we can write ψ ( x ) = −
12 ( x + · · · + x s ) + √− ϕ ( x )for some real linear function ϕ ( x ). Thus, for for multi-indices I ⊂ [ s ] , K, L ⊂ [ t = 1], wehave Ψ IK ¯ L ( x ) = P i ∈ I x i ( K = L = ø) P i ∈ I x i − P i ∈ ˇ I x i + √− ϕ ( x ) ( K = { } , L = ø) P i ∈ I x i − P i ∈ ˇ I x i − √− ϕ ( x ) ( K = ø , L = { } ) − P i ∈ ˇ I x i ( K = { } , L = { } ) . We can say that E e Ψ IK ¯ L ( x ) = E e Ψ I ′ K ′ ¯ L ′ ( x ) if and only if ( I, J, K ) = ( I ′ , J ′ , K ′ ), ( I, J, K ) =(ø , ø , ø) and ( I ′ , J ′ , K ′ ) = ([ s ] , { } , { } ) or ( I, J, K ) = ([ s ] , { } , { } ) and ( I ′ , J ′ , K ′ ) =(ø , ø , ø).We compute H p,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) for each I, K, L so that E e Ψ IK ¯ L ( x ) is non-trivial i.e.( I, K, L ) = (ø , ø , ø) , ([ s ] , { } , { } ). If K = L = ø, H p,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) = (cid:26) h α I i ∧ V q h ¯ α , . . . , ¯ α s i ( p = | I | )0 (otherwise) . If K = { } , L = ø, H p,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) = (cid:26) h α I ∧ β i ∧ V q h ¯ α , . . . , ¯ α s i ( p = | I | + 1)0 (otherwise) . If K = ø , L = { } , H p,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) = (cid:26) h α I i ∧ V q − h ¯ α , . . . , ¯ α s i ∧ h ¯ β i ( p = | I | )0 (otherwise) . If K = { } , L = { } , H p,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) = (cid:26) h α I ∧ β i ∧ V q − h ¯ α , . . . , ¯ α s i ∧ h ¯ β i ( p = | I | + 1)0 (otherwise) . In particular for any
I, K, L with (
I, K, L ) = (ø , ø , ø) , ([ s ] , { } , { } ), H ,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) =0. Remark 5.5.
In this case, the equality dim H p,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) = (cid:18) sq − | L | (cid:19) holds (see the inequality as in Corollary 5.1). INE BUNDLES AND HODGE SYMMETRY ON OELJEKLAUS-TOMA MANIFOLDS 11
As noted in Remark 3.2, we haveΘ ∼ = E e x ⊕ · · · ⊕ E e xs ⊕ E e ψ x ) ⊕ · · · ⊕ E e ψt ( x ) . Hence, for 0 < p ≤ s + 1, we have H ,q ( M, p ^ Θ) ∼ = M | I | + | K | = p,L =ø H ,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) = 0 . This means the following statement.
Proposition 5.6.
For an OT-manifold X ( K, U ) with t = 1 , we have H ,q ( X ( K, U ) , p ^ Θ) = 0 . Since H , ( X ( K, U ) , Θ) = 0, every OT-manifold X ( K, U ) with t = 1 is rigid (cf. [1]).Moreover by H , ( X ( K, U ) , V Θ) = 0, X ( K, U ) with t = 1 does not admit non-zero holo-morphic Poisson structure. This implies that on every OT-manifold X ( K, U ) with t = 1,small deformations of generalized complex structures are given by B -field transformations(see [3]). Example 5.7 ([11]) . In [11, Section 3.1], Otiman gives a field K and a subgroup U ⊂ O ∗ + K such that s = t = 2 and for any u ∈ U , σ ( u ) σ ( u ) σ ( u ) = σ ( u ) σ ( u ) σ ( u ) = 1. By theserelations, we can write ψ ( x ) = − x + √− ϕ ( x ) , ψ ( x ) = − x + √− ϕ ( x )for some real linear functions ϕ ( x ) , ϕ ( x ). Thus, for examples, we compute H p,q (Γ \ G ) = V q h ¯ α , ¯ α i ( p = 0) h α ∧ β ∧ ¯ β , α ∧ β ∧ ¯ β i ∧ V q − h ¯ α , ¯ α i ( p = 2 , q ≥ h α ∧ α ∧ β ∧ β ∧ ¯ β ∧ ¯ β i ∧ V q − h ¯ α , ¯ α i ( p = 4 , q ≥ H p,q (Γ \ G, E e x ) = h α i ∧ V q h ¯ α , ¯ α i ( p = 1) h α ∧ α ∧ β ∧ ¯ β i V q − h ¯ α , ¯ α i ( p = 3 , q ≥ . Hence, for I = K = J = { } and q ≥
2, we have H ,q (Γ \ G, E e Ψ IK ¯ L ( x ) ) = H ,q (Γ \ G ) = 2 (cid:18) q − (cid:19) > (cid:18) q − (cid:19) . References [1] D. Angella, M. Parton, V. Vuletescu, Rigidity of Oeljeklaus-Toma manifolds. arXiv:1610.04045 toappear in Ann. Inst. Fourier[2] K. Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures. ForumMath. (2000), no. 1, 77–96.[3] M. Gualtieri, Generalized complex geometry. Ann. of Math. (2) (2011), no. 1, 75–123.[4] N. Istrati, A. Otiman, De Rham and twisted cohomology of Oeljeklaus-Toma manifolds. Ann.Inst. Fourier (Grenoble) (2019), no. 5, 2037–2066.[5] H. Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds. Bull. Lond. Math.Soc. (2013), no. 1, 15–26. [6] H. Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with localsystems. J. Differential Geom. (2013), no. 2, 269–297.[7] H. Kasuya, de Rham and Dolbeault cohomology of solvmanifolds with local systems. Math. Res.Lett. (2014), no. 4, 781–805.[8] H. Kasuya, Remarks on Dolbeault cohomology of Oeljeklaus-Toma manifolds and Hodge theory.arXiv:2008.06649 To appear in Proc. Amer. Math. Soc[9] K. Oeljeklaus, M. Toma, Non-K¨ahler compact complex manifolds associated to number fields.Ann. Inst. Fourier (Grenoble) (2005), no. 1, 161–171.[10] A. Otiman, M. Toma, Hodge decomposition for Cousin groups and Oeljeklaus-Toma manifolds,arXiv: 1811.02541, to appear in Annali della Scuola Normale di Pisa[11] A. Otiman, Special Hermitian metrics on Oeljeklaus-Toma manifolds, arXiv:2009.02599[12] M. S. Raghunathan, Discrete subgroups of Lie Groups, Springer-Verlag, New York, 1972. Department of Mathematics, Graduate School of Science, Osaka University, Osaka,Japan
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