Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows
aa r X i v : . [ m a t h . DG ] F e b GRIFFITHS POSITIVITY FOR BISMUT CURVATURE ANDITS BEHAVIOUR ALONG HERMITIAN CURVATUREFLOWS
GIUSEPPE BARBARO
Abstract.
In this note we study a positivity notion for the curvatureof the Bismut connection; more precisely, we study the notion of
Bismut-Griffiths-positivity for complex Hermitian non-K¨ahler manifolds. Sincethe K¨ahler-Ricci flow preserves and regularizes the usual Griffiths posi-tivity we investigate the behaviour of the Bismut-Griffiths-positivity un-der the action of the Hermitian curvature flows. In particular we studytwo concrete classes of examples, namely, linear Hopf manifolds andsix-dimensional Calabi-Yau solvmanifolds with holomorphically-trivialcanonical bundle. From these examples we identify some HCFs whichdo not preserve Bismut-Griffiths-non-negativity. Introduction
In this note we introduce a notion of positivity (or non-negativity) forcomplex Hermitian manifolds which emulates the definition of Griffiths pos-itivity (non-negativity) and refers to the Bismut curvature tensor. Then weinvestigate its behaviour under the action of the Hermitian curvature flows.Let us now introduce our problem giving more details. The Bismut con-nection ∇ + on a Hermitian manifold ( M, g, J ) is the unique Hermitian con-nection with totally skew-symmetric torsion. It can be defined by the for-mula g ( ∇ + X Y, Z ) = g ( ∇ LCX
Y, Z ) + 12
J dω ( X, Y, Z )where ∇ LC is the Levi-Civita connection, ω is the canonical 2-form associ-ated to g and J acts as J dω ( · , · , · ) = − dw ( J · , J · , J · ).We define a notion of positivity emulating the definition of the Griffithspositivity for the Chern connection of the tangent bundle (also known asholomorphic bisectional curvature). We do this by evaluating the holomor-phic Bismut bisectional curvature of the Hermitian manifold. Definition.
A Hermitian manifold ( M, g, J ) has Bismut-Griffiths-positive (resp. non-negative ) curvature if its Bismut curvature tensor Ω B satisfies • Ω B ∈ ∧ , M ⊗ ∧ , M ; Key words and phrases.
Hermitian Curvature Flows; Bismut connection; holomorphicbisectional curvature; linear Hopf manifolds; six-dimensional Calabi-Yau solvmanifolds. • for any non-zero ξ, ν ∈ T , M Ω B ( ξ, ξ, ν, ν ) > resp. ≥ . The condition Ω ∈ ∧ , M ⊗ ∧ , M is known in the literature as (Cplx).We note that the curvature tensor Ω associated to a Hermitian connectionsatisfies (Cplx) if and only if it satisfies the J -invariance formula: Ω ijkl = 0.We could define our notion of positivity even if (Cplx) were not satisfied;however, in that case we would describe only the geometry of the (1 ,
1) partof Ω B ignoring the (2 ,
0) and (0 ,
2) components.
Remark.
Given a complex Hermitian manifold ( M, g, J ) , if the metric ispluriclosed, the Bismut curvature tensor (in special coordinate around z )become: Ω Bijkl = Ω
Chklij − g pq T kpj T lqi Thus for SKT manifolds the Bismut-Griffiths positivity (non-negativity) im-plies the Griffiths positivity (non-negativity).In [15] F. Tong studies a positivity notion for the tensor Ω Chklij − g pq T kpj T lqi which arises naturally from a Bochner-type formula for closed (1 , -forms. We are interested in the behaviour of the Bismut-Griffiths positivity underthe action of the
Hermitian Curvature Flows (HCFs). In the article [14],Streets and Tian suggest this class of flows as a new class of parabolic flowsof metrics on Hermitian manifolds, and proved short time existence andregularity results. There are examples in the literature of the use of theseflows to reveal information about the structure of M as a complex manifold,see for example the works of Streets and Tian on the Pluriclosed flow [12],[13] and Ustinovskiy [20], [22] and [21]. In particular, we are motivated bythe possibility of detecting some regularization properties as in [22]. In thatarticle Ustinovskiy showed that there is a flow in the HCF family (whichwe will call
Ustinovskiy flow ) that not only preserves Griffiths positivityand non-negativity of the Chern connection, but it evolves a metric withnon-negative Griffiths curvature everywhere and positive in some point to ametric with positive Griffiths curvature everywhere.The HCFs are defined by the equation ∂∂t g = − S + Q where S is the trace of the Chern curvature tensor Ω Ch S ij = ( T r ω Ω Ch ) ij = g kl Ω Chklij and Q is a quadratic polynomial in the torsion T Ch of the Chern connection.More precisely, the components of the quadratic term Q are: Q ij = g kl g mn T Chikn T Chjlm Q ij = g kl g mn T Chkmj T Chlni Q ij = g kl g mn T Chikl T Chjnm Q ij = 12 g kl g mn (cid:16) T Chmkl T Chnji + T Chmij T Chnlk (cid:17)
ISMUT-GRIFFITHS POSITIVITY AND HCF 3
Studying the notion of Bismut-Griffith positivity, we focus on six dimen-sional Calabi-Yau solvmanifolds. These are compact quotients of solvableLie groups endowed with invariant complex structures and with holomorphi-cally trivial canonical bundle. In this class we found examples of manifoldswhich do not satisfy (Cplx) (see Theorem 1) and among that which satisfy(Cplx) we found Bismuth-Griffiths-non-negative manifolds. We also provethat
Theorem (Theorem 3) . Let M be a six-dimensional Calabi-Yau solvman-ifold, then any Hermitian curvature flow preserves (Cplx). Moreover, Her-mitian curvature flows also preserve Bismut-Griffiths-non-negativity on thesemanifolds. We also study our problem on linear Hopf manifolds. The linear Hopfsurface with its standard metric ( g H , see §
3) has flat Bismut curvature, thusit is our first example of Bismut-Griffiths-non-negative manifold. We alsofind other (non flat) examples of metrics with Bismut-Griffiths-non-negativecurvature on linear Hopf manifolds of higher dimensions.Our results come from the analysis of a class of canonical metrics ( g ( α, β )metrics, see § g H on linear Hopf manifolds. We prove Theorem (Proposition 1 & Corollary 1) . On a linear Hopf manifold equippedwith a metric g ( α, β ) the Bismut curvature tensor satisfies (Cplx). More-over, on these manifolds any HCF starting from g ( α, β ) preserves (Cplx). We characterize the metrics in this family which have Bismut-Griffiths-non-negative curvature ( § Theorem (Theorem 5 & Proposition 4) . There exists a class in the familyHCF of flows which do not preserve Bismut-Griffiths-non-negativity; all theothers HCFs preserve Bismut-Griffiths-non-negativity on linear Hopf man-ifolds equipped with g ( α, β ) metrics. Finally, we check these conditions on some interesting Hermitian curva-ture flows, such as the Ustinovskiy flow and the pluriclosed and the gradient flows of Streets and Tian (the latter is the only HCF which is a gradient flowfor some functional F , see [14]). In particular, the Ustinovskiy flow does notpreserve the Bismut-Griffiths-non-negativity, while the gradient flow doeson the linear Hopf manifolds with g ( α, β ) metrics; the pluriclosed flows pre-serves the Bismut-Griffiths-non-negativity on the Hopf surfaces while it doesnot on linear Hopf manifolds of higher dimension. See § GIUSEPPE BARBARO Bismut-Griffiths-positivity of 6-dimensional Calabi-Yausolvmanifolds
In this section we analyze the symmetries of (Cplx) and the notion ofBismut-Griffiths-positivity by investigating them on 6-dimensional Calabi-Yau solvmanifolds. By solvmanifold we mean a compact quotient of con-nected simply-connected solvable Lie group by a co-compact discrete sub-group. We endow it with a Hermitian structure ( g, J ) which is invariant(under left-translations) when lifted to the universal cover; moreover, weask these solvmanifolds to be Calabi-Yau, that is, the complex structure J is such that the canonical bundle is holomorphically-trivial. This includesnilmanifolds with invariant complex structure.We refer to the classification (up to linear equivalence) of the invariant com-plex structures on six-dimensional nilmanifolds (Table 1) and solvmanifoldsnon-nilmanifolds with holomorphically-trivial canonical bundle (Table 2) asoutlined in the works of Salamon, Ugarte, Villacampa, Andrada, Barberis,Dotti, Ceballos and Otal [11], [17], [1], [19], [4]. Name Complex structure Lie algebra (Np) dϕ = dϕ = 0 , dϕ = ρ ϕ , where ρ ∈ { , } ρ = 0 : h = (0 , , , , , ρ = 1 : h = (0 , , , ,
13 + 42 ,
14 + 23)(Ni) dϕ = dϕ = 0, h = (0 , , , , , h = (0 , , , , ,
12 + 34) dϕ = ρ ϕ + ϕ + λ ϕ + D ϕ , h = (0 , , , , ,
14 + 23) h = (0 , , , ,
13 + 42 ,
14 + 23)where ρ ∈ { , } , λ ∈ R ≥ , D ∈ C with ℑ D ≥ h = (0 , , , , , h = (0 , , , , , dϕ = 0 , dϕ = ϕ , h = (0 , , , , , h = (0 , , , , ,
14 + 25) h = (0 , , , , , dϕ = ρϕ + B ϕ + c ϕ , h = (0 , , , , ,
14 + 23) h = (0 , , , , , ρ ∈ { , } , B ∈ C , c ∈ R ≥ , with ( ρ, B, c ) = (0 , , h = (0 , , , ,
13 + 14 , h = (0 , , , , ,
13 + 42) h = (0 , , , ,
13 + 42 ,
14 + 23) h = (0 , , , , , dϕ = 0 , dϕ = ϕ + ϕ , h − = (0 , , , , , − dϕ = √− ρ ϕ ± √− ϕ − ϕ ) , where ρ ∈ { , } h +26 = (0 , , , , ,
14 + 25)
Table 1.
Invariant complex structures on six-dimensionalnilmanifolds up to linear equivalence, see [1], [4], [19].
ISMUT-GRIFFITHS POSITIVITY AND HCF 5
Name Complex structure Lie algebra (Si) dϕ = Aϕ + Aϕ , g = (15 , − , − , , ,
0) when θ = 0 dϕ = − Aϕ − Aϕ , dϕ = 0, g α = ( α ×
15 + 25 , −
15 + α × , − α ×
35 + 45 , − − α × , , A = cos θ + √− θ,θ ∈ [0 ,π ) with α = cos θ sin θ ≥
0, when θ = 0(Sii) dϕ = 0 , dϕ = − ϕ − (cid:0) + √− x (cid:1) ϕ + √− xϕ , g = (0 , − , , , − , − dϕ = ϕ + (cid:16) − √− x (cid:17) ϕ + √− x ϕ ,where x ∈ R > (Siii1) dϕ = √− ϕ + √− ϕ g = (23 , − , , − , , dϕ = −√− ϕ − √− ϕ dϕ = ± ϕ (Siii2) dϕ = ϕ + ϕ g = (24 + 35 , , , − , − , dϕ = − ϕ − ϕ dϕ = ϕ + ϕ (Siii3) dϕ = √− ϕ + √− ϕ g = (24 + 35 , − , , − , , dϕ = −√− ϕ − √− ϕ dϕ = ϕ + ϕ (Siii4) dϕ = √− ϕ + √− ϕ g = (24 + 35 , , , − , − , dϕ = −√− ϕ − √− ϕ dϕ = ± ( ϕ − ϕ )(Siv1) dϕ = − ϕ , dϕ = ϕ , dϕ = 0 g = (16 − ,
15 + 26 , −
36 + 45 , − − , , dϕ = 2 √− ϕ + ϕ , x ∈ { , } dϕ = − √− ϕ + xϕ , dϕ = 0(Siv3) dϕ = Aϕ − ϕ dϕ = − Aϕ + ϕ , dϕ = 0 A ∈ C with | A |6 = 1(Sv) dϕ = − ϕ g = (45 ,
15 + 36 , −
26 + 56 , − , , dϕ = √− ϕ + ϕ − √− ϕ dϕ = − √− ϕ + √− ϕ Table 2.
Invariant complex structures on six-dimensionalsolvmanifolds non-nilmanifolds with holomorphically-trivialcanonical bundle up to linear equivalence, see [9], [5].In the formulas above the authors refer to a co-frame ( ϕ , ϕ , ϕ , ϕ , ϕ , ϕ )where ( ϕ , ϕ , ϕ ) is an invariant co-frame of (1 , J .The generic invariant Hermitian structure ω = g ( J · , · ) is given by(1)2 ω = √− r ϕ + s ϕ + t ϕ ) + uϕ − uϕ + vϕ − vϕ + zϕ − zϕ where ϕ ij = ϕ i ∧ ϕ j and the coefficients satisfy the following inequalitiescoming from the fact that g is positive definite (see [17]): r > , s > , t > r s > | u | , r t > | z | , s t > | v | √− r s t + 2 Re ( √− uvz ) − ( r | v | + t | u | + s | z | ) > GIUSEPPE BARBARO where, Ξ denotes the Hermitian matrix associated to the Hermitian struc-ture, i.e. Ξ = √− r u z − u √− s v − z − v √− t Analyzing case by case the possible families of nilmanifolds and solv-manifolds, we get the following results, whose proofs are collected in theAppendix.
Theorem 1.
Let M be a six-dimensional solvmanifold endowed with in-variant metric g and complex structure J , g as in (1) and J such that thecanonical bundle is holomorphically-trivial. The Bismut curvature tensorsatisfies the (Cplx) condition precisely in the cases (Np), (Ni), (Nii), (Si),(Siii1), (Siv1) and (Siv3) when the conditions on the invariant structures ofTable 3 are satisfied. Name (Cplx) condition Bismut-Griffiths-non-negativity (Np) Always satisfied nowhere non-negative nor non-positive(Ni) h : non-negative if u = 0 h , D = 1: non-negative h , D = −
1: nowhere non-negative nor non-positive ρ = 0 h : nowhere non-negative nor non-positive h : nowhere non-negative nor non-positive h : non-negative(Nii) c = B = 0, ρ = 1, v = 0 nowhere non-negative nor non-positive(Si) u = v = z = 0 A = √−
1: flat A = √−
1: nowhere non-negative nor non-positive(Siii1) u = v = z = 0 non-negative(Siv1) Always satisfied nowhere non-negative nor non-positive(Siv3) u = v = z = 0 nowhere non-negative nor non-positive A = 0, v = z = 0 in both cases Table 3.
Conditions on the underlying complex structure,invariant Hermitian metric and Lie algebras.
Remark.
In [2] the authors studied the existence of Gauduchon K¨ahler-likeconnections on 6-dimensional Calabi-Yau solvmanifolds. First of all, theGauduchon connections are an affine line of Hermitian connections whichgoes through the Chern and the Bismut connections. Moreover, K¨ahler-likemeans that the curvature tensor satisfies both (Cplx) and the first Bianchiidentity. Examples on the Hopf manifolds show that the K¨ahler-like condi-tion is strictly stronger than (Cplx).
ISMUT-GRIFFITHS POSITIVITY AND HCF 7
In the cases where (Cplx) is satisfied we look at the Bismut-Griffithspositivity.
Theorem 2.
Let M be a six-dimensional solvmanifold endowed with in-variant metric g and complex structure J , g as in (1) and J such that thecanonical bundle is holomorphically-trivial. If J is in the families (Siii1) or(Ni) with Lie algebra h , h and h (with D = 1 ) then the Bismut curvaturetensor satisfies (Cplx) and it is Bismut-Griffiths non-negative. If J is in thefamily (Si) with Lie algebra g and diagonal metric the manifold is Bismut-flat. In all the other cases where (Cplx) is satisfied the invariant metrics areneither non-positive nor non-negative. (See Table 3) The computations (in the Appendix) lead to the Remarks 1, 2, 3, 4 and5 that we summarize in the following statement.
Theorem 3.
Let M be a six-dimensional solvmanifold endowed with invari-ant metric g and complex structure J , g as in (1) and J such that the canon-ical bundle is holomorphically-trivial. Then the symmetries of (Cplx) arepreserved by any HCF. Moreover, the HCFs also preserve Bismut-Griffiths-non-negativity and Bismut-flatness, when they occur. We remark that recent results in [6] show that the pluriclosed flow (whichis in the HCF family) preserves the Bismut K¨ahler-like condition on 2-stepnilpotent Lie group with left-invariant Hermitian structure.All the computations on six-dimensional Calabi-Yau solvmanifolds withinthe proofs of the above statements are contained in the Appendix.3.
HCFs on Linear Hopf Manifolds
Linear Hopf manifolds are defined as quotients of the complex domain C n \ { } over an equivalence relation depending on α ∈ C n \ { } M n = C n \ { }∼ , where ( z , · · · , z n ) ∼ ( α z , · · · , α n z n ) with | α | = · · · = | α n |6 = 1.These manifolds come with a natural complex structure and a Hermitianmetric ω H = √− δ ij | z | dz i ∧ dz j . The Bismut curvature tensor associated to g H satisfies various symmetries,including (Cplx). Indeed, its non-vanishing coefficients are (see [7]):Ω Bijkl ( z ) = δ il δ jk − δ ij δ kl | z | + δ ij z k z l + δ kl z i z j − δ il z j z k − δ jk z i z l | z | . Thus, for any ξ, ν ∈ T , M Ω B ( ξ, ξ, ν, ν ) | z = 1 | z | ( − | ξ | | ν | | z | + | ξ · ν | | z | + | ν · z | | ξ | + | z · ξ | | ν | + − ( ξ · ν )( ν · z )( z · ξ ) − ( ν · ξ )( ξ · z )( z · ν )) . GIUSEPPE BARBARO
Since this vanishes for n = 2, we get the following Proposition.
For n = 2 the Hopf manifold with the canonical metric g H isBismut flat. Thus, in particular, it is Bismut-Griffiths-non-negative. Proposition.
The Hopf manifold with the canonical metric g H is not Bismut-Griffiths non-negative for n > .Proof. The Bismut curvature tensor satisfies Ω
Bijkl = − Ω Bkjil = − Ω Bilkj . ThusΩ
Bijkl = Ω
Bklij and the two Bismut-Ricci curvatures agree and are R Bij = 1 | z | (cid:0) (2 − n )( δ ij | z | − z i z j ) (cid:1) , so that for any ξ ∈ T , MR B ( ξ, ξ ) = 2 − n | z | ( | ξ | | z | −| ξ · z | ) ≤ , and the equality holds only if ξ = λz with λ ∈ C .Thus, if we take ξ different from the multiples of z > R Bz ( ξ, ξ ) = g ij Ω Bz ( ξ, ξ, ∂ i , ∂ j ) = | z | X i Ω Bz ( ξ, ξ, ∂ i , ∂ i ) , so at least one of the Ω Bz ( ξ, ξ, ∂ i , ∂ i ) is strictly negative. (cid:3) HCF-closed family of metrics on linear Hopf manifolds.
Herewe study the Hermitian metrics g ( α, β ) (depending on real parameters α > β > − α ) on a generic linear Hopf manifold defined by g ( α, β ) ij = α δ ij | z | + β z i z j | z | . This class of metrics is useful for our problem, in fact it is closed by theaction of the HCFs (see Proposition 3). Since g H = g (1 ,
0) this familynaturally arises studying the evolution of HCFs on linear Hopf manifolds.As a matter of fact, these metrics also arise in the evolution of g H by theChern-Ricci flow (see [16]). We also recall that, for similar reasons, the g ( α, β ) metrics also appear in [8] where the authors use them to produceexamples of Levi-Civita Ricci-flat Hermitian metrics on Hopf manifolds.The inverse of g ( α, β ) is g ( α, β ) ij = | z | α (cid:18) δ ij − βα + β z j z i | z | (cid:19) . The Christoffel symbols for the Bismut connection areΓ kij = g ks ∂ j g is = 1 | z | (cid:18) βα δ kj z i − δ ki z j (cid:19) − βα z i z j z k | z | ;Γ kij = g ks (cid:16) ∂ i g js − ∂ s g ji (cid:17) = 1 | z | (cid:18) δ ij z k − α + βα δ kj z i (cid:19) + βα z i z j z k | z | , ISMUT-GRIFFITHS POSITIVITY AND HCF 9 and a direct computation leads to Ω
Bijkl = 0 for any i, j, k, l ∈ { , . . . , n } .Thus we have the following, Proposition 1.
The linear Hopf manifolds equipped with a generic Hermit-ian metric g ( α, β ) satisfy (Cplx). Now we turn to the study of Bismut-Griffiths-positivity for these metrics.We have the following formula for the Bismut curvature tensor of a g ( α, β )-metric:Ω Bijkl = α (cid:20) δ il δ jk − δ ij δ kl | z | + δ ij z k z l + δ kl z i z j − δ il z j z k − δ jk z i z l | z | (cid:21)| {z } U α + 2 β (cid:20) − δ ij δ kl | z | + δ ij z k z l + δ kl z i z j | z | + − z i z j z k z l | z | (cid:21)| {z } U β . Notice that we already know U α since it is the curvature tensor of g H (w.r.t.the Bismut connection). If we evaluate U β on vectors ξ, ν ∈ T , M , we get U β ( ξ, ξ, ν, ν ) = 1 | z | ( | ξ | | z | −| ξ · z | )( | ν · z | −| ν | | z | ) ≤ , (2)and the equality holds if and only if ξ = λz or ν = λz with λ ∈ C .In dimension two U α vanishes, thus g ( α, β ) is Bismut-Griffiths-non-negativeif and only if β ≤
0. In dimension greater than 2, we have seen that U α isnot non-negative, as well as U β ; hence the non-negativity of the tensor willdepend only on the ratio γ . = βα . In particular we get Lemma 1.
For any dimension n , the Hermitian metric g ( α, − α ) on then-dimensional linear Hopf manifold is Bismut-Griffiths-non-negative; in par-ticular, given any ξ, ν ∈ T , M , we have the following equation Ω B ( ξ, ξ, ν, ν ) = α | z | (cid:12)(cid:12) ( ξ · ν ) | z | − ( ξ · z )( z · ν ) (cid:12)(cid:12) ≥ . Remark.
Notice that in any point z ∈ M we will never get Ω B ( z ) > formetrics g ( α, β ) , since both the terms U α and U β vanish if ξ = λz or ν = λz for λ ∈ C . Proposition 2.
There are coefficients γ n depending on the dimension n of the linear Hopf manifold such that for all α > , g ( α, γα ) has Bismut-Griffiths-non-negative curvature tensor if and only if γ ≤ γ n . This coeffi-cients are γ = 0 and γ n = − for n ≥ .Proof. We already know that γ = 0 has the above property.Now suppose that n ≥
3. Take α > ε >
0; by Lemma 1 we knowthat the metric g ( α, − α ) is Bismut-Griffiths-non-negative and the metric g ( α, ( − + ε ) α ) has Bismut curvature tensor given byΩ B ( ξ, ξ, ν, ν ) = α | z | (cid:12)(cid:12) ( ξ · ν ) | z | − ( ξ · z )( z · ν ) (cid:12)(cid:12) + 2 εU β ( ξ, ξ, ν, ν ) . On a point z ∈ M with two zero coordinates (say k and l ), by equation (2)we get Ω B ( ∂ k , ∂ k , ∂ l , ∂ l ) z = 2 εU β ( ∂ k , ∂ k , ∂ l , ∂ l ) z = − ε | z | < . (cid:3) HCFs evolution in the canonical family.
First of all, we computethe terms S and Q of the HCFs in the explicit case of a linear Hopf manifoldequipped with g ( α, β ) metric. By direct computations we get the Christoffelsymbols of the Chern connectionΓ kij = 1 | z | (cid:18) βα δ ki z j − δ kj z i (cid:19) − βα z i z j z k | z | , and the Chern curvaturesΩ lijk = 1 | z | (cid:20) δ lk (cid:18) δ ij − z i z j | z | (cid:19) − βα δ li (cid:18) δ jk − z k z j | z | (cid:19) + βα ( δ jk z i + δ ij z k ) | z | − z i z j z k | z | z l (cid:21) ;Θ (2) ij = 1 | z | (cid:20)(cid:18) n − − βα (cid:19) δ ij + βα (cid:18) n − βα ( n − (cid:19) z i z j | z | (cid:21) . The Chern torsion is T kij = | z | (cid:16) βα + 1 (cid:17) ( δ ki z j − δ kj z i ) and we have the fol-lowing quadratic terms in T Ch : Q ij = 1 | z | (cid:18) βα + 1 (cid:19) (cid:20) αα + β δ ij + (cid:18) n − βα + β (cid:19) z i z j | z | (cid:21) ; Q ij = 2 | z | (cid:18) βα + 1 (cid:19) αα + β (cid:20) δ ij − z i z j | z | (cid:21) ; Q ij = ( n − (cid:18) βα + 1 (cid:19) z i z j | z | ; Q ij = 1 | z | (cid:18) βα + 1 (cid:19) αα + β ( n − (cid:20) δ ij − z i z j | z | (cid:21) . Thus any metric g ( α, β ) evolving along the HCF (for any choice of Q ) re-mains a linear combination of δ ij | z | and z i z j | z | at any times. Proposition 3.
Let M be a linear Hopf manifold equipped with the Her-mitian metric g ( α , β ) , then the HCF ˙ g = − S + aQ + bQ + cQ + dQ starting from g ( α , β ) evolves the metric as g ( t ) ij = α ( t ) δ ij | z | + β ( t ) z i z j | z | for t ≥ , where α and β satisfy the ODE system (3) α (0) = α , β (0) = β ˙ α ( t ) = βα + 1 − n + (cid:16) βα + 1 (cid:17) ( a + 2 b + ( n − d )˙ β ( t ) = βα (cid:16) − n − βα ( n − (cid:17) + (cid:16) βα + 1 (cid:17) ( n − a + ( n − c )+ − (cid:16) βα + 1 (cid:17) ( a + 2 b + ( n − d ) Corollary 1.
On a linear Hopf manifold (of any dimension) equipped witha metric g ( α , β ) any HCF starting from g ( α , β ) preserves (Cplx). Thus we now turn to the study of the behaviour of the Bismut-Griffiths-non-negativity on linear Hopf manifolds equipped with metrics in the fam-ily g ( α, β ) under the action of the HCFs. Since the Bismut-Griffiths-non-negativity of the g ( α, β )-metrics only depends on the ratio γ , we give a usefulresult (Theorem 4) which describes the evolution of γ through the action ofthe HCFs.Notice that in the ODE (3) both ˙ α and ˙ β depends only on γ = βα . Thisis natural since the flows act the same way on homothetic metrics. Thus wecan rewrite the ODE system as α (0) = α , β (0) = β ˙ α ( t ) = γ + 1 − n + ( γ + 1) ( a + 2 b + ( n − d )˙ β ( t ) = γ (1 − n − γ ( n − γ + 1) ( n − a + ( n − c )+ − ( γ + 1) ( a + 2 b + ( n − d )Then the ratio γ evolves as(4) γ (0) = β α ˙ γ = α ( γ + 1) [( F − n ) γ + F ]˙ α = ( γ + 1) L − n where F ( a, b, c, d, n ) = ( n − a − b + ( n − c − ( n − d and L ( a, b, c, d ) = 1 + a + 2 b + ( n − d .Recall that β > − α , so γ > −
1. Thus γ can not be equal to − Fn − F only if F < n . If we start with α , β with ratio Fn − F , then α and β will evolve as two straight lines.In particular, when γ = Fn − F the flow acts on the metric by homotheties:these metrics are called static. Moreover, the following result shows thatthese static metrics are globally stable in the g ( α, β )-family for the HCFs. Theorem 4.
On an n -dimensional linear Hopf manifold consider the HCF ˙ g = − S + aQ + bQ + cQ + dQ . Suppose that the coefficients ( a, b, c, d ) aresuch that F ( a, b, c, d, n ) < n , then the metric g (1 , Fn − F ) (as well as any ofits multiples) is static for the flow. Moreover, any metric g ( α , β ) evolvesalong the flow so that the ratio γ converges to Fn − F . Proof.
First of all, notice that the flows act by homothety on the staticmetrics, thus they are exactly those for which˙ γ = 1 α ( γ + 1) [( F − n ) γ + F ] = 0 . Since γ > −
1, this could be the case only if
F < n and we get γ = Fn − F .Now suppose that the starting metric g ( α , β ) has ratio γ < Fn − F . Bythe evolution equation (4) for γ we know that γ is strictly increasing alongthe flow, moreover, it is bounded above from Fn − F . We now distinguish twocases, depending on L : when α is decreasing along the flow, meaning that( Fn − F + 1) L − n ≤
0, and when it is not.In the first case, suppose that γ does not converge to Fn − F , then it needs toconverge to some γ ∞ with γ < γ ∞ < Fn − F . In this way˙ α < ( γ ∞ + 1) L − n < (cid:18) Fn − F + 1 (cid:19) L − n ≤ , thus ˙ α is uniformly strictly negative and so α will get to zero in finite time,say T ; at the same time T , γ will be increasing with infinite speed (by eq.(4)), which is a contradiction to the convergence γ → γ ∞ .In the second case, namely (cid:16) Fn − F + 1 (cid:17) L − n >
0, we can suppose withoutloss of generality that ( γ + 1) L − n >
0. Moreover, since the term ( γ + 1)in the formula of ˙ γ is positive and increasing we can suppress it and provethe convergence of γ to Fn − F with evolution equation˙ γ = 1 α [( F − n ) γ + F ] . Since γ is bounded from above, then also ˙ α is so. This means that we canbound α above with a straight line with positive slope α ≤ α (0) + At . Thusfinally we have ( α (0) + At ) ˙ γ ( t ) ≥ ( F − n ) γ ( t ) + F . We have an explicitsolution for this ODE γ ( t ) ≥ C ( At + α (0)) F − nA + Fn − F , where the constant C depends on the initial value γ . Since the exponent F − nA is negative we get the convergence to Fn − F for t → ∞ .A similar argument holds true also in the opposite case, namely if γ > Fn − F . (cid:3) This is another evidence of stability results as in [14]. Here we have globalstability in the non-KE setting; however, this is an extremely particular casesince it refers to linear Hopf manifolds and the starting point needs to be ametric of the type g ( α, β ). ISMUT-GRIFFITHS POSITIVITY AND HCF 13
Bismut-Griffith-non-negativity under the action of HCFs.
Inthis section we detect a subset of HCF of flows which preserve Bismut-Griffiths-non-negativity on linear Hopf manifolds with g ( α, β ) metrics (seeTheorem 5). This subfamily is prescribed by inequalities of the coefficients( a, b, c, d ) characterizing the HCFs which depend on the dimension n . Con-sider γ n as in Proposition 2. Theorem 5.
Consider an n -dimensional linear Hopf manifold equipped withmetrics of type g ( α , β ) , and suppose that ( n − a − b +( n − c − ( n − d ≤ n γ n γ n +1 . Then if the metric g ( α , β ) is Bismut-Griffiths-non-negative, theHCF with coefficients ( a, b, c, d ) starting at g ( α , β ) preserves the Bismut-Griffiths-non-negativity.Proof. Notice that since the metric g ( α , β ) is Bismut-Griffiths-non-negative,the initial ratio γ must be γ ≤ γ n . Moreover, we have that F ( a, b, c, d, n ) = ( n − a − b + ( n − c − ( n − d ≤ n γ n γ n + 1 ≤ < n thus by Theorem 4 the ratio γ will evolve along the flow converging to avalue γ ∞ = Fn − F ≤ γ n . This means that the metric will remain Bismut-Griffiths-non-negative along the flow. (cid:3) On the other hand, if the above inequality is not satisfied then the flowdoes not preserve Bismut-Griffiths-non-negativity. More precisely
Proposition 4.
On linear Hopf manifolds of dimension n equipped withmetrics of type g ( α, β ) , if ( n − a − b + ( n − c − ( n − d > n γ n γ n +1 thenthe HCF with coefficients ( a, b, c, d ) does not preserve Bismut-Griffiths-non-negativity.Proof. To prove the statement we suppose to perform the HCF with coef-ficients ( a, b, c, d ) starting from the metric g (1 , γ n ). By hypothesis this isBismut-Griffiths-non-negative and γ n is the largest ratio for which this hap-pens; thus we just have to verify that ˙ γ (0) > F ≥ n , we get that ( F − n ) γ + F > γ since γ > − n > F ( a, b, c, d, n ) = ( n − a − b + ( n − c − ( n − d > n γ n γ n + 1then ( F − n ) γ n + F >
0. Thus, if the inequality in the statement is satisfied,using the evolution equation (4) for γ we get that˙ γ (0) = 1 α ( γ n + 1)[( F − n ) γ n + F ] > (cid:3) Remark.
Proposition 4 shows that the inequalities of Theorem 5 are sharp,meaning that they detect the largest set of HCFs which preserve Bismut-Griffiths-non-negativity on linear Hopf manifolds equipped with metrics inthe family g ( α, β ) . Interesting flows.
We can use the above set of inequalities to checkif some interesting flow preserve Bismut-Griffiths-non-negativity on linearHopf manifolds with metrics g ( α, β ).The Gradient flow of Streets and Tian, that is the one with coefficients a = , b = − , c = − , d = 1, has Fn − F = − n − n +1 ; hence, Fn − F ≤ − ≤ γ n for any n > Fn − F = − when n = 2. On the other hand, for theUstinovskiy flow F = 1 > pluriclosed flow , introducedby Streets and Tian in [13]. They identified a particular choice of Q whichyields a flow that preserves the pluriclosed condition and so has a naturallink with the Bismut connection. Specifically, in our notation Q is identifiedby a = 1 , b = c = d = 0. With this coefficients we get F = n −
2. Thuscomparing these values with the inequalities in the previous results we have
Proposition.
Consider a linear Hopf manifold (of any dimension) equippedwith a Bismut-Griffiths-non-negative metrics g ( α , β ) , then • the gradient flow of Streets and Tian starting at g ( α , β ) evolvespreserving the Bismut-Griffiths-non-negativity; • the Ustinovskiy flow starting at g ( α , β ) does not preserve the Bismut-Griffiths-non-negativity; • unless n = 2 , the pluriclosed flow starting at g ( α , β ) does not pre-serve the Bismut-Griffiths-non-negativity. Remark.
We saw that the pluriclosed flow performed on linear Hopf sur-faces with g ( α, β ) metrics preserves the Bismut-Griffiths-non-negativity. Asa matter of fact, a metric g ( α, β ) on the linear Hopf manifold is pluriclosedif and only if n = 2 ; thus it is interesting that the pluriclosed flow behavewell with Bismut-Griffiths-non-negativity only in dimension two. Appendix. Computations on six-dimensional Calabi-Yausolvmanifolds
We collect here the computations on Calabi-Yau solvmanifolds that leadto Theorems 1, 2 and 3.Some of the following computation were performed with the help of thesymbolic computation software Sage [10].4.1.
Nilmanifolds.
Holomorphically-parallelizable nilmanifolds in Family (Np).
Considersix-dimensional holomorphically-parallelizable nilmanifolds, i.e. nilmani-folds with holomorphically trivial tangent bundle. On these nilmanifoldsthe complex structure equations are dϕ = dϕ = 0 , dϕ = ρϕ ; ρ = 0 , ρ = 0 refers to the Torus and is K¨ahler and flat; thus we consideronly the case of Iwasawa manifold ( ρ = 1). ISMUT-GRIFFITHS POSITIVITY AND HCF 15
A direct calculation shows that Ω ij ·· = Ω ·· kl = 0 for any i, j, k, l ∈ { , , } .Moreover, we have computed the following determinant and coefficient:Ω Ω − Ω Ω = − t √− = t ( r t − | z | )16 √− Nilmanifolds in Family (Ni).
Consider the generic Hermitian struc-ture of this family dϕ = dϕ = 0; dϕ = ρϕ + ϕ + λϕ + Dϕ where ρ ∈ { , } , λ ≥ ImD ≥ v = z = 0 and r = 1 in the generic expression (1):2 ω = √− ϕ + s ϕ + t ϕ ) + uϕ − uϕ Let us take the element Ω = ρs t √− . It vanishes if and only if ρ = 0.Taking into account the classification of complex structures up to equivalence(see [4]) we set the coefficients ρ, λ and D (and the Lie algebras) as follows: • ( ρ, λ, D ) = (0 , , √− h ; • ( ρ, λ, D ) = (0 , , ± h ; • ( ρ, λ, D ) = (0 , , ), Lie algebra h ; • ( ρ, λ ) = (0 ,
1) and D ∈ [0 , ), Lie algebra h ; • ( ρ, λ, D ) = (0 , , h .In any of these cases the computation of the curvature elements yields that,for any i, j, k, l ∈ { , , } Ω ij ·· = Ω ·· kl = 0.Suppose λ = 0 (thus Lie algebras h , h and h ). From direct computationswe get the following elements of the Bismut curvature tensorΩ = t Ω = Re ( D ) t Ω = Re ( D ) t Ω = | D | t Ω = − Re ( √− D )( s − | u | ) t u Ω = − Re ( √− D )( s − | u | ) t u Thus if D = − h ), then Ω < < Ω and the curvature tensor isneither non-negative nor non-positive. On the other hand, if D = √− h ),then the determinantΩ Ω − Ω Ω = − t | u | ( s − | u | ) ≤ . Thus the Bismut curvature tensor is non-negative if and only if u = 0.Finally, for D = 1 or D = 0 ( h or h ) we have Bismut-Griffiths non-negativity. Remark 1.
Suppose we are in case of Lie algebra h , hence with coefficients ( ρ, λ, D ) = (0 , , √− . Suppose also that u = 0 (i.e. the metric g isdiagonal, since we are supposing v = z = 0 ) then also S and Q are diagonal.This means that any HCF preserves the condition u = v = z = 0 . Now we turn to the Lie algebras h and h , for which we have computedthe following element and the determinant of the curvature tensor:Ω = t > , Ω Ω − Ω Ω = t ( D −
14 ) . Thus, in case of Lie algebra h (i.e. D < ) the Bismut curvature tensoris neither non-positive nor non-negative. Now, if D = , we compute thesecond Ricci tensors of The Bismut curvature Ric B . We have that Ric B Ric B − Ric B Ric B = − | s − √− u + 1 | s − | u | ) < . Nilmanifolds in Family (Nii).
Consider the complex structure equa-tions dϕ = 0 , dϕ = ϕ , dϕ = ρϕ + Bϕ + cϕ where ρ ∈ { , } , c ≥ B ∈ C satisfying ( ρ, B, c ) = (0 , , ρ = 0 we have the coefficientsΩ = c ( s t − | v | ) √− , Ω = − B ( s t − | v | ) √− c = B = 0 which is impossible since ( ρ, B, c ) = 0. Thuswe take ρ = 1 and computeΩ = s t − | v | √− t B .
Hence also B = 0.Now we prove that c = v = 0. First of all, if c = 0, we haveΩ = s t − | v | √− v , which implies v = 0; on the other hand, if v = 0, we haveΩ = ct r t − | z | √− , which implies c = 0. Thus c = 0 if and only if v = 0. Suppose c = 0 (hence v = 0), then we compute the following elements of the Bismut curvature ISMUT-GRIFFITHS POSITIVITY AND HCF 17 tensor:Ω = (cid:2) √− ct ( r t z + √− t | u | + uvz − uvz − z | z | ) −√− t uvz − ( cv + v ) | z | v (cid:3) / √− = (cid:2) √− ct ( s t u − u | v | − v | z | + r t v − √− s vz ) −√− t uv − ( cv + v ) v z (cid:3) / √− = (cid:2) ct ( − √− r | v | −√− s z + s uv − √− uvz )+( √− s vz − u | v | )( cv + v ) (cid:3) / √− = t √− u | v | + s vz − ct uz − √− cr t v √− = Ω = 0 we get: (cid:2) √− ct ( r t − | z | ) − ( √− t u + vz )( cv + v ) (cid:3) z = ct u ( t u − √− vz ) (cid:2) √− ct ( r t − | z | ) − ( √− t u + vz )( cv + v ) (cid:3) v = − ct s ( √− t u + vz )Hence ( √− t u + vz )( uv + √− s z ) = 0 . Notice that from this equations we also get that u = 0 if and only if z = 0;however they can not vanish or we would get cv = 0 from Ω = 0. Thus u, z ( v, c ) are different from zero and we distinguish two cases: √− t u + vz = 0 and uv + √− s z = 0. In the first case, we have0 = √− ct ( r t − | z | ) − ( √− t u + vz )( cv + v )= √− ct ( r t − | z | ) , thus c = 0, which is a contradiction. In the second case, Ω = 0 and uv + √− s z = 0 imply0 = √− u | v | + s vz = ct ( uz + √− r v ) . Multiplying by v and using again uv + √− s z = 0 we obtain s | z | = r | v | .Finally, with these equations Ω becomeΩ = − ct = 0 . This shows that v = c = 0 is needed to satisfy (Cplx).With these parameters ( ρ = 1; c = B = 0) and v = 0 (Cplx) is satisfiedand we get the following element and determinant of the Bismut curvaturetensor: Ω = s t √− Ω − Ω Ω = − t √− showing that the curvature tensor is neither non-negative nor non-positive. Remark 2.
With parameters ( ρ = 1; c = B = 0 ) the condition v = 0 (andhence (Cplx)) is preserved by any Hermitian curvature flow in the familyHCF. Nilmanifolds in Family (Niii).
Consider the complex structure equa-tions dϕ = 0 , dϕ = ϕ + ϕ , dϕ = √− ρϕ + δ √− ϕ − ϕ )where ρ ∈ { , } and δ = ±
1. From a direct computation we get thefollowing elements of the Bismut curvature tensor:Ω = − ( √− uv + s z ) v √− = − ( √− δρs z − √− r v − δρuv − uz ) s v √− = ( √− t u + zv ) s v √− = √− s z − s uvz + s uvz − √− u v + √− v | u | √− u, v and z must vanish: suppose v = 0, then impos-ing Ω = 0 we get s z = −√− uv . Now Ω = 0 implies r v = √− uz ,and Ω = 0 implies t u = √− vz . These three equations together wouldimply that det Ξ = 0 which is a contradiction, thus v must vanish. Moreover,if v = 0 from Ω = 0 we get also z = 0. Finally, Ω with v = z = 0 isΩ = − s t ( √− s − t ) u √− . Thus, also u mast vanish.Now, for u = v = z = 0 we have Ω = ( δ √− t − s ) = 0, showing that(Cplx) is never satisfied.4.2. Solvmanifolds.
Solvmanifolds in Family (Si).
Consider the generic Hermitian struc-ture of this family dϕ = A ( ϕ + ϕ ) , dϕ = − A ( ϕ + ϕ ) , dϕ = 0where A = cos θ + √− θ and θ ∈ [0 , π ).We directly compute (and set equal to zero)Ω = −√− | A | r uv + s z u √− . ISMUT-GRIFFITHS POSITIVITY AND HCF 19
This vanishes only if r uv + s z u = 0 since A = 0. We compute thefollowing coefficients of the Bismut curvature tensor (where I = √− = − (cid:2) Ar t | u | +( A + A ) r | v | − ( A + 3 A ) Ir zuv +( A + A ) Ir uvz − ( A − A ) | u | | z | (cid:3) A/ √− = − (cid:2) − IAr s t u − ( A + A ) r s zv + ( A − A ) Ir | v | u +(3 A − A ) Is | z | u − ( A − A ) u vz (cid:3) A/ √− = (cid:2) − IAr s t u + ( A + A ) r s vz + (3 A − A ) Ir | v | u +( A − A ) Is | z | u + ( A − A ) zu v (cid:3) A/ √− = − (cid:2) As t | u | +( A + A ) s | z | − ( A + A ) Is zuv +(3 A + A ) Is uvz − ( A − A ) | u | | v | (cid:3) A/ √− u = v = z = 0 as unique solution. The computations follow exactly thesame structure as for solvmanifolds in family (Siv3), see § Remark 3.
The invariant metric g with u = v = z = 0 is Chern-flat. More-over, with these parameters also Q is diagonal; hence (Cplx) is preserved byany HCF. We computed the following elements of the Bismut curvature tensor:Ω = 2 Re ( A ) r t , Ω = − r Re ( A ) , showing that if A = √− A = √−
1, corresponding to the Lie algebra g , the diagonalmetrics are K¨ahler, hence, K¨ahler-flat (see the Remark above). By [3],the complex solvmanifold is in fact biholomorphic to a holomorphically-parallelizable manifold.4.2.2. Solvmanifolds in Family (Sii).
Consider the complex structure equa-tions (where x ∈ R > ) dϕ = 0 dϕ = − ϕ − ( 12 + √− x ) ϕ + √− xϕ ,dϕ = 12 ϕ + ( 12 − √− x ) ϕ + √− x ϕ Working on the elements Ω and Ω (which we set equal to zero) weget s = t , see [2] for details. ThenΩ = t (2 x − √− x = 0 , and so (Cplx) is never satisfied.4.2.3. Solvmanifolds in Families (Siii1), (Siii3), (Siii4).
Recall that the Liealgebras underlying (Siii1), (Siii3), and (Siii4) are, respectively, g , g , and g . In order to give a unified argument, we will gather the complex structureequations as follows: dϕ = √− ϕ + ϕ ) , dϕ = −√− ϕ + ϕ ) , dϕ = xϕ + yϕ where ( x, y ) = ( ± ,
0) for g , ( x, y ) = (1 ,
1) for g and ( x , y = − x ) =( ± , ∓
1) for g . In particular x = 0.Imposing the symmetries (Cplx) on the Bismut curvature tensor, we getthat y must be zero (meaning that the underling Lie algebra is g ) and themetric described by the equation (1) need to satisfy u = v = z = 0; see [2]for details.With these condition (Cplx) is satisfied and the only non-zero coefficients oftype Ω ijkl of the curvature tensor is Ω = t . Remark 4. If u = v = z = 0 (i.e. the metric g is diagonal) then also S and Q are diagonal. This means that any HCF preserves the condition u = v = z = 0 . Solvmanifolds in Family (Siii2).
The complex structure equations forthis family are the following: dϕ = ϕ + ϕ , dϕ = − ϕ − ϕ , dϕ = ϕ + ϕ Imposing the symmetries (Cplx) on the Bismut curvature tensor, we getthat the metric described by the equation (1) need to satisfy v = z = 0; see[2] for details.From a direct computation we getΩ = √− r t u t + 2 √− u √− , Ω = r s t t + 2 √− u √− = 0 implies t + 2 √− u = 0, and then Ω = 0 leads to t + 2 √− u .These two equations together imply that u is real which is in contradictionwith both of them. This shows that (Cplx) is never satisfied.4.2.5. Solvmanifolds in Families (Siv1).
Consider the complex structure equa-tions for this family: dϕ = − ϕ , dϕ = ϕ , dϕ = 0A direct computation shows that Ω ij ·· = Ω ·· kl = 0 for any i, j, k, l ∈ { , , } .Moreover, we have the following coefficient and determinant of the curvaturetensor:Ω = r s − | u | √− r , Ω Ω − Ω Ω = − r s − | u | √− r Solvmanifolds in Families (Siv2).
Recall the complex structure equa-tions for this family: dϕ = 2 √− ϕ + ϕ , dϕ = − √− ϕ − xϕ , dϕ = 0where x = 0 , and Ω :Ω = − ( r s − | u | )( xr s + x | u | +2 √− r u )8 det ΞΩ = − ( r s − | u | )( r s + | u | − √− xs u )8 det ΞNotice that, Ω = Ω = 0 if and only if xr s + x | u | +2 √− r u = r s + | u | − √− xs u = 0 . If x = 1, these equations imply Re ( u ) = 0, Im ( u ) = − r and r = s , whichis a contradiction to the positive definiteness of the metric. Hence, x = 0and Ω is always different from zero.4.2.7. Solvmanifolds in Families (Siv3).
The complex structure equationsfor this family are the following (with A ∈ C \ S ): dϕ = Aϕ − ϕ , dϕ = − Aϕ + ϕ , dϕ = 0We directly compute (and set equal to zero)Ω = √− r uv + s z u ) A √− . This vanishes if A = 0 or r uv + s z u = 0. We start analyzing the case A = 0. We compute the following coefficients of the Bismut curvature tensor(where I = √− = (cid:2) Ar t | u | +( A − r | v | +(1 − A ) Ir zuv +( A − Ir uvz − ( A + 1) | u | | z | (cid:3) √− = (cid:2) − IAr s t u + (3 A + 1) Is u | z | +(1 + A ) Ir u | v | +(1 − A ) r s zv − ( A + 1) u vz (cid:3) / √− = (cid:2) IAr s t u + (1 − A ) r s vz − (3 A + 1) Ir | v | u − ( A + 1) Is | z | u − ( A + 1) zu v (cid:3) / √− = (cid:2) As t | u | +( A − s | z | +(1 − A ) Is zuv +(3 A − Is uvz − ( A + 1) | u | | v | (cid:3) / √− A − = 0 by hypothesis, thus if u = 0 we get also v = 0 and z = 0 from Ω = 0 and Ω = 0 respectively. On the other hand, if u = 0 then v vanishes if and only if z vanishes (from r uv + s z u = 0), and they can not vanish together otherwise u should also be 0 (from Ω = 0).Now suppose u, v, z = 0 and consider the following equations:Ω − Ω = s | z | − r | v | √− A √− | v | − Ω | z | = − uvz + uvz √− A √− r | v | = 0Ω uv − Ω uz = A | u | s t ( √− r v − uz ) − √− r | v | v √− uv − Ω uz = A | u | r t ( uv − √− s z ) − √− r | v | z √− · z − (6) · v = 0 we get vz = 0 which is a contradiction.This shows that u, v and z must be zero and a direct computation showsthat with this hypothesis (Cplx) is satisfied.In case A = 0, Ω and Ω becomeΩ = ( r s − | u | )( √− r v + zu )16 √− , Ω = ( r s − | u | )( uv − √− s z )16 √− √− r v + zu = 0 and √− s z − uv = 0 implies that v vanishes if and only if z vanishes. Moreover, if they are both different fromzero, we can multiply the first one by v and the second one by z ; this leadsto √− r | v | + uvz = 0 = √− s | z | − uvz which is impossible. Hence v and z must be zero and with this hypothesis (Cplx) is satisfied. Remark 5.
In both cases u = v = z = 0 ; A = 0 and v = z = A = 0 theseconditions (and then (Cplx)) are preserved by any Hermitian curvature flowin the family HCF. Now, setting v = z = 0, we get the following elements of the curvaturetensor: Ω = 12 r t ( A − A −
1) ;Ω = −
12 ( A − A − r s − (( A − A − A − r | u | r s − | u | , showing that in both cases u = 0 and A = 0 the curvature tensor is neithernon-negative nor non-positive.4.2.8. Solvmanifolds in Families (Sv).
Recall the complex structure equa-tions for this family: dϕ = − ϕ , dϕ = √− ϕ + 12 ϕ − √− ϕ , dϕ = − √− ϕ + √− ϕ Consider the terms Ω and Ω : if we setΩ = s | z | √− , we get z = 0, but then Ω = − r s −| u | t = 0; thus (Cplx) is never satisfied. Acknowledgements
I would like to thank my advisor Daniele Angella for many helpful sugges-tions and for his constant support and encouragement. I am also gratefulto Simone Diverio and Luis Ugarte Vilumbrales for their comments andsuggestions.
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Dipartimento di Matematica “Guido Castelnuovo”, Universit`a la Sapienza,Piazzale Aldo Moro, 5, 00185 Roma, Italy
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