Pluriclosed and Strominger Kähler-like metrics compatible with abelian complex structures
aa r X i v : . [ m a t h . DG ] F e b PLURICLOSED AND STROMINGER K ¨AHLER-LIKE METRICSCOMPATIBLE WITH ABELIAN COMPLEX STRUCTURES
ANNA FINO, NICOLETTA TARDINI, AND LUIGI VEZZONI
Abstract.
We show that the existence of a left-invariant pluriclosed Hermitian metricon a unimodular Lie group with a left-invariant abelian complex structure forces thegroup to be 2-step nilpotent. Moreover, we prove that the pluriclosed flow starting froma left-invariant Hermitian metric on a 2-step nilpotent Lie group preserves the StromingerK¨ahler-like condition. Introduction
A Hermitian metric g on a complex manifold ( M, J ) is called pluriclosed (or
SKT ) ifits fundamental form ω ( · , · ) = g ( J · , · ) satisfies(1) dJ dω = 0 . The pluriclosed condition (1) can be characterized in terms of the torsion of the Bismut (orStrominger) connection ∇ B . Indeed, in [10] Bismut proved that on a Hermitian manifold( M, J, g ) there is a unique Hermitian connection ∇ B whose torsion T B , once regarded as a(3 , g , is skew-symmetric. The pluriclosed condition is equivalent to dT B = 0.If T B = 0, the Bismut connection ∇ B coincides withe the Levi-Civita condition and themetric g is K¨ahler.By [30] a Hermitian metric g is pluriclosed and satisfies the condition ∇ B T B = 0 if andonly if its Bismut curvature R B satisfies the first Bianchi identity(2) σ x,y,z R B ( x, y, z ) = 0and the type condition(3) R B ( x, y, z ) = R B ( J x, J y, z ) , for any tangent vectors x, y, z in M . Hermitian metrics satisfying (2) and (3) are calledin literature Strominger K¨ahler-like and have been studied recently in [5, 30, 29, 17].An important tool in the geometry of pluriclosed metrics is the so-called pluriclosed flow which is a parobolic flow of Hermitian metrics which preserves the pluriclosed condition[26, 27]. A natural question is to see if the Strominger K¨ahler-like condition is preservedby the flow.Every conformal class of any Hermitian metric on a compact complex surface admits apluriclosed metric, but in higher dimensions, the existence of a pluriclosed metric is notautomatically guaranteed anymore. Looking at the existence of left-invariant pluriclosed
Mathematics Subject Classification.
Key words and phrases. pluriclosed metric, unimodular Lie algebra, abelian complex structure, pluri-closed flow. metrics on 6-dimensional nilpotent Lie groups endowed with a left-invariant complexstructure, only 4 out of the 34 isomorphism classes admit pluriclosed metrics and they areall two-step nilpotent, leading to the question whether this is a general feature in arbitrarydimensions [18]. It turns out that two of the 4 classes in dimension six admit StromingerK¨ahler-like metrics [5] and that the complex structure is abelian. More in general, acharacterization of 2-step nilpotent Lie algebras admitting Strominger K¨ahler-like metricshave been obtained in [31], showing in particular that the left-invariant complex structurehas to be abelian.We recall that a left-invariant complex structure on a real Lie group G of real dimension2 n is completely determined by a complex structure J on the Lie algebra g of G , i.e. byan endomorphism satisfying J = − Id and the integrability condition J [ x, y ] − [ J x, y ] − [ x, J y ] − J [ J x, J y ] = 0 , ∀ x, y ∈ g . The complex structure J is called abelian if(4) [ J x, J y ] = [ x, y ] , ∀ x, y ∈ g . or equivalently if the i -eigenspace of J , denoted with g , , is an abelian subalgebra of g C := g ⊗ R C (that motivates the terminology introduced in [8]). By [23] a Lie algebraadmitting an abelian complex structure has abelian commutator, thus, it is 2-step solvable.Recent results about the existence of pluriclosed metrics on solvable Lie groups havebeen obtained in [22, 14, 7, 15, 19].The purpose of this paper is twofold. On one hand we study the existence of a pluri-closed metric on a unimodular Lie group with an abelian complex structure and on theother hand we investigate the interplay between the Strominger K¨ahler-like conditionand the pluriclosed flow. We recall that a Lie group G is unimodular if and only if | det ( Ad g ) | = 1, for every g ∈ G , where Ad is the adjoint representation. For a connectedLie group G this is equivalent to requiring that tr ( ad X ) = 0, for every X ∈ g , where g isthe Lie algebra of G .The existence of other types of Hermitian inner products compatible with abelian com-plex structures, like for instance K¨ahler [3], balanced [4] and locally conformally K¨ahler inner products [4], has been already studied in letterature. In [13] the second authorand the third author, in collaboration with H. Kasuya, proved that on non-abelian Liealgebras with an abelian complex structure there are no Hermitian-symplectic structures.The latters can be regarded as special pluriclosed inner products and the natural follow-up is focusing on the existence of pluriclosed metrics compatible with abelian complexstructures.Our first result is the following
Theorem 1.1.
Let g be a unimodular Lie algebra with an abelian complex structure J .If ( g , J ) admits a pluriclosed inner product, then g is -step nilpotent. In the particular case when the commutator of g is totally real the result follows from[19, Corollary 5.7], but our proof does not make use of the argument in [19]. Moreover,Theorem 1.1 generalizes [13, Proposition 6.1]. LURICLOSED AND SKL METRICS COMPATIBLE WITH ABELIAN COMPLEX STRUCTURES 3
Next we focus on the existence of Strominger K¨ahler-like metrics in relation to thepluriclosed flow. By using the characterization in [31] of left-invariant Strominger K¨ahler-like metrics on 2-step nilpotent Lie groups, we prove the following
Theorem 1.2.
Let ( G, J, g ) be a -step nilpotent Lie group with a left-invariant Stro-minger K¨ahler-like Hermitian structure and let g t be the solution to the pluriclosed flowstarting from g . Then g t is Strominger K¨ahler-like for every t .Acknowledgements. The paper is supported by Project PRIN 2017 “Real and com-plex manifolds: Topology, Geometry and Holomorphic Dynamics” and by GNSAGA ofINdAM. 2.
Proof of Theorem 1.1
We first need the following
Lemma 2.1.
Let g be a Lie algebra with an abelian complex structure J and an Hermitianinner product g . Then, the torsion -form T B of the Bismut connection of ( g , J, g ) satisfies T B ( x, y, z ) = − g ([ x, y ] , z ) − g ([ y, z ] , x ) − g ([ z, x ] , y ) , for every x, y, z, w ∈ g .Proof. Let ω be the fundamental form of g . Let x, y, z, w ∈ g , then T B ( x, y, z ) = − dω ( J x, J y, J z ) and we directly compute dω ( J x, J y, J z ) = − ω ([ J x, J y ] , J z ) − ω ([ J y, J z ] , J x ) − ω ([ J z, J x ] , J y )= − ω ([ x, y ] , J z ) − ω ([ y, z ] , J x ) − ω ([ z, x ] , J y ) . Hence the claim follows. (cid:3)
As a consequence we have
Proposition 2.2.
Let ( g , J ) be a Lie algebra with an abelian complex structure. A Her-mitian inner product g on ( g , J ) is pluriclosed if and only if (5) g ([ y, z ] , [ w, x ]) − g ([ x, z ] , [ w, y ]) + g ([ x, y ] , [ w, z ]) = 0 for every x, y, z, w ∈ g .Proof. We recall that g is pluriclosed if and only if dT B = 0. Let x, y, z, w ∈ g , then, bythe previous Lemma, dT B ( w, x, y, z ) = − T B ([ w, x ] , y, z ) + T B ([ w, y ] , x, z ) − T B ([ w, z ] , x, y ) − T B ([ x, y ] , w, z ) + T B ([ x, z ] , w, y ) − T B ([ y, z ] , w, x )= g ([[ w, x ] , y ] , z ) + g ([ y, z ] , [ w, x ]) + g ([ z, [ w, x ]] , y ) − g ([[ w, y ] , x ] , z ) − g ([ x, z ] , [ w, y ]) − g ([ z, [ w, y ]] , x )+ g ([[ w, z ] , x ] , y ) + g ([ x, y ] , [ w, z ]) + g ([ y, [ w, z ]] , x )+ g ([[ x, y ] , w ] , z ) + g ([ w, z ] , [ x, y ]) + g ([ z, [ x, y ]] , w ) − g ([[ x, z ] , w ] , y ) − g ([ w, y ] , [ x, z ]) − g ([ y, [ x, z ]] , w )+ g ([[ y, z ] , w ] , x ) + g ([ w, x ] , [ y, z ]) + g ([ x, [ y, z ]] , w )= 2( g ([ y, z ] , [ w, x ]) − g ([ x, z ] , [ w, y ]) + g ([ x, y ] , [ w, z ])) , ANNA FINO, NICOLETTA TARDINI, AND LUIGI VEZZONI where, in the last equality, we used the Jacobi identity.Therefore, g is pluriclosed if and only if g ([ y, z ] , [ w, x ]) − g ([ x, z ] , [ w, y ]) + g ([ x, y ] , [ w, z ]) = 0 , as required. (cid:3) Remark 2.3.
Note that from the complex point of view, condition (5) is equivalent to g ([ z , ¯ z ] , [ z , ¯ z ]) = g ([ z , ¯ z ] , [ z , ¯ z ]) , for every z , z , z , z ∈ g , .From now on, for a Lie algebra g with an abelian complex structure J we will denoteby ζ the center of g and by g J the ideal g J = [ g , g ] + J [ g , g ] . Note that g J is a J -invariant Lie subalgebra of g .Under the hypothesis of Proposition 2.2 we obtain the following characterization in termsof the center ζ of g . Corollary 2.4.
Let ( g , J, g ) be a Lie algebra with an abelian complex structure and apluriclosed inner product. Then k [ x, y ] k + k [ x, J y ] k = g ([ x, J x ] , [ y, J y ]) for every x, y ∈ g . In particular, x ∈ g lies in the center of g if and only if [ x, J x ] = 0 , i.e., ζ = { x ∈ g : [ x, J x ] = 0 } . Proof.
By using (5) for x, y ∈ g we have k [ x, y ] k + k [ x, J y ] k = g ([ x, y ] , [ x, y ]) + g ([ x, J y ] , [ x, J y ])= g ([ J x, J y ] , [ x, y ]) − g ([ J x, y ] , [ x, J y ]) = g ([ y, J y ] , [ x, J x ]) , and the claim follows. (cid:3) We will need the following
Lemma 2.5.
Let g be a unimodular Lie algebra with an abelian complex structure J .Then, g J = g . Proof.
By contradiction, assume that g J = g . Then, since by hypothesis g is an abelianideal in g , by [9, Proposition 4.1] ( g /ζ , J ) is holomorphically isomorphic to aff ( A ) forsome commutative algebra A . Since, g is unimodular, also g /ζ is unimodular, and so aff ( A ) is unimodular. So, by [4, Lemma 2.6], A is nilpotent and aff ( A ) is a nilpotent Liealgebra. As a consequence, we have that g /ζ is also nilpotent implying that g is nilpotenttoo. But, this is absurd since by [25] for a nilpotent Lie algebra g we have g J = g . (cid:3) Proposition 2.6.
Let g be a Lie algebra with an abelian complex structure J . Assumethat ( g , J ) has a pluriclosed inner product g and g J is -step nilpotent. Then g is -stepnilpotent. LURICLOSED AND SKL METRICS COMPATIBLE WITH ABELIAN COMPLEX STRUCTURES 5
Proof.
Write g = ( g J ) ⊥ ⊕ g J with respect to the inner product g . Since g J is nilpotent and has a pluriclosed innerproduct, its center u is J -invariant. We write g J = u ⊥ ⊕ u The key observation is that u is contained in the center of g . Indeed, if x ∈ u , then inparticular we have [ x, J x ] = 0 and Corollary 2.4 implies that x belongs to the center of g .Now let f ∈ ( g J ) ⊥ . We show that [ f, x ] lies in the center of g , for every x ∈ g .Set D := ad f : g J → g J . Since J is abelian, ad f J = − ad Jf and therefore, D and DJ are both derivations. Moreover, D [ x, y ] = 0 for every x, y ∈ g J ; indeed from the 2-stepnilpotency of g J , we have that [ x, y ] ∈ u , for every x, y ∈ g J , and that u ⊂ ζ .Let x ∈ g J and y ∈ g . Then, taking into account that g is 2-step solvable, Corollary2.4 yields that k [ Dx, y ] k + k [ Dx, J y ] k = g ([ Dx, J Dx ] , [ y, J y ]) = g ([ DJ x, Dx ] , [ y, J y ]) = 0 , from which we deduce that [ f, x ] is in the center of g for all x ∈ g J .Now let f , f ∈ ( g J ) ⊥ . By Jacobi identity[[ f , f ] , x ] = 0for every x ∈ g J . Hence [ f , f ] ∈ u and so in the center of g , as required. (cid:3) Now we are ready to prove Theorem 1.1.
Proof of Theorem . . We work by induction on the complex dimension n of g . The basecase n = 1 is trivial and we assume that the statement holds up to complex dimension n −
1. Let ( g , J, g ) be a Lie algebra of complex dimension n with an abelian complexstructure and a pluriclosed inner product. In view of Lemma 2.5, g J is a proper Liesubalgebra and inherits an abelian complex structure and a pluriclosed inner product.By induction assumption g J is 2-step nilpotent. Hence Proposition 2.6 implies that g is2-step nilpotent and the claim follows. (cid:3) Remark 2.7.
By Theorem 1.1, if g is a unimodular Lie algebra with an abelian complexstructure J and a pluriclosed inner product, then g is 2-step nilpotent. In particular,notice that g J is abelian. Indeed, since J is abelian, g is 2-step solvable and[ g , g ] = [ J g , J g ] = 0 . Moreover from the 2-step nilpotency of g we infer that also[ g , J g ] = 0 . As a consequence, if X = Γ \ G is a nilmanifold endowed with an invariant abelian complexstructure J and a pluriclosed metric g , then by [18, Theorem A], X is a total space of aprincipal holomorphic torus bundle over a torus. ANNA FINO, NICOLETTA TARDINI, AND LUIGI VEZZONI
Notice that from Theorem 1.1 in particular follows that a nilpotent Lie algebra with anabelian complex structure and admitting a pluriclosed inner product is necessarily 2-step.This partially confirm the conjecture that the existence of a pluriclosed inner product ona nilpotent Lie algebra g with a complex structure forces g to be 2-step.Moreover, it is quite natural wondering how rigid is the existence of other kind ofspecial inner products on a Lie algebra with a complex structure. In particular, the so-called astheno-K¨ahler metrics introduced by Jost and Yau in [20], which are characterizedby the condition ∂∂ω n − = 0 . Clearly, on a complex surface any Hermitian metric is astheno-K¨ahler and in complexdimension 3 the notion of astheno-K¨ahler metric coincides with that of pluriclosed. Herewe observe that in the nilpotent case the existence of a astheno-K¨ahler inner producton a Lie algebra compatible with an abelian complex structure does not force the 2-stepcondition in contrast to Theorem 1.1 for the pluriclosed case.
Example 2.8.
In view of [21, Corollary 5.1.9] we consider the 8-dimensional 3-step nilpo-tent Lie algebra g with complex structure equations dϕ = dϕ = 0 , dϕ = ϕ , dϕ = B ϕ + B ( ϕ + ϕ ) + D ( ϕ + ϕ ) , with D = 0. In particular, the complex structure J is abelian.Let ω = X k =1 ix k ¯ k ϕ k ¯ k + X ≤ k Let G be a 2-step nilpotent Lie group with a left-invariant Hermitian structure ( g, J )and denote by g its Lie algebra. Assume further that g is pluriclosed. In view of [31], themetric g is Strominger K¨ahler-like if and only if there exists an orthonormal basis { x i } si =1 of g = [ g , g ] and an orthonormal basis { ǫ i } ni =1 of g such that1. J ǫ i = ǫ i + n , i = 1 , . . . , n ;2. g + J g = span { ǫ r +1 , . . . , ǫ n , ǫ n + r +1 , . . . , ǫ n } ;3. the only non-trivial brackets under { ǫ i } are[ ǫ i , ǫ n + i ] = λ i x i , i = 1 , . . . , s, for some positive numbers { λ i } si =1 and n − r ≤ s ≤ min { r, n − r ) } .Note, that in particular J has to be abelian.If { ǫ i } is the dual basis to { ǫ i } , then the metric g writes as g = n X k =1 ( ǫ k ) . LURICLOSED AND SKL METRICS COMPATIBLE WITH ABELIAN COMPLEX STRUCTURES 7 From [31] it follows that every other left-invariant pluriclosed metric h taking the diagonalform h = n X k =1 a k ǫ k ǫ k , a k > , for every k = 1 , . . . , n , is Strominger K¨ahler-like since we can modify the basis { ǫ k } to˜ ǫ k = 1 √ a k ǫ k which still satisfies items 1.,2.,3..Moreover, in view of [12], the Ricci form of the Bismut connection of h takes thefollowing expression ρ Bh ( x, y ) = 12 s X k =1 a k h ([ ǫ k , ǫ k + n ] , [ x, y ]) . which implies that ρ Bh takes the diagonal form ρ Bh = s X k =1 b k ǫ k ∧ ǫ n + k . It follows that, by uniqueness, the solution to the pluriclosed flow starting from g isdiagonal for every t and the claim of Theorem 1.2 follows. Remark 3.1. Notice that we can give a more explicit expression for the Ricci form ρ Bg t of the Bismut connection ∇ B of the metric g t . Let { ǫ i } be a basis satisfying items 1.,2.,3.and g = X ( ǫ k ) . Consider the solution to the pluriclosed flow g t = X a tk ( ǫ k ) . If { ǫ tk } is a g t -orthonormal basis satisfying items 1.,2.,3., namely ǫ tk = 1 p a tk ǫ k , then ρ Bg t ( x, y ) = 12 X g t ([ ǫ tk , ǫ tn + k ] , [ x, y ]) . We have [ ǫ tk , ǫ tn + k ] = 1 p a tk p a tn + k [ ǫ k , ǫ n + k ] = 1 a tk λ k x k and [ ǫ tk , ǫ tn + k ] = λ tk x tk with { x tk } g t -orthonormal.Hence ρ Bg t ( x, y ) = 12 X λ tk g t ( x tk , [ x, y ]) . ANNA FINO, NICOLETTA TARDINI, AND LUIGI VEZZONI Now ρ Bg t ( ǫ i , ǫ n + i ) = 12 s X k =1 λ tk g t ( x tk , [ ǫ i , ǫ n + i ]) . Since [ ǫ i , ǫ n + i ] = a ti [ ǫ ti , ǫ tn + i ] = a i λ ti x ti we get ρ Bg t ( ǫ i , ǫ n + i ) = 12 s X k =1 λ tk g t ( x tk , [ ǫ i , ǫ n + i ]) = 12 s X k =1 λ tk a ti g t ( x tk , λ ti x ti ) = 12 ( λ ti ) a ti . Therefore ρ Bg t = 12 s X k =1 ( λ tk ) a tk ǫ k ∧ ǫ n + k . References [1] A. Andrada, M. L. Barberis, I. Dotti, Classification of abelian complex structures on 6-dimensionalLie algebras, J. Lond. Math. 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Zheng, Strominger connection and pluriclosed metrics, arXiv:1904.06604 .[31] Q. Zhao, F. Zheng, Complex nilmanifolds and K¨ahler-like connections, J. Geom. Phys. Dipartimento di Matematica “G. Peano”, Universit`a degli studi di Torino,Via Carlo Alberto 10, 10123 Torino, Italy Email address : [email protected] (Nicoletta Tardini) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unit`adi Matematica e Informatica, Universit`a degli Studi di Parma, Parco Area delle Scienze53/A, 43124 Parma, Italy Email address : [email protected] (Luigi Vezzoni) Dipartimento di Matematica “G. Peano”, Universit`a degli studi di Torino,Via Carlo Alberto 10, 10123 Torino, Italy Email address ::