Abstract
We study Hermitian metrics whose Bismut connection \(\nabla ^B\) satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition \(R^B(x,y,z,w)=R^B(Jx,Jy,z,w)\), for every tangent vectors x, y, z, w, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in Angella et al. (Commun Anal Geom, to appear, 2018), Yau et al. (2019) and Zhao and Zheng (2019). Using the characterization of SKT almost abelian Lie groups in Arroyo and Lafuente (Proc Lond Math Soc (3) 119:266–289, 2019), we construct new examples of Hermitian manifolds satisfying the Bismut Kähler-like condition. Moreover, we prove some results in relation to the pluriclosed flow on complex surfaces and on almost abelian Lie groups. In particular, we show that, if the initial metric has constant scalar curvature, then the pluriclosed flow preserves the Vaisman condition on complex surfaces.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Given a Hermitian manifold (X, J, g), the Bismut connection \(\nabla ^B\) is the unique connection on X that is Hermitian (i.e. such that \(\nabla ^B g =0\) and \(\nabla ^B J =0\)) and has totally skew-symmetric torsion tensor (cf. [8]). If the torsion 3-form of \(\nabla ^B\) is closed, the Hermitian metric g is called SKT or pluriclosed. Examples of SKT manifolds are given by Lie groups (and its compact quotients) endowed with left-invariant SKT Hermitian structures (see for instance [11, 12, 24]). A characterization of almost abelian Lie groups, i.e. Lie groups whose Lie algebra has a codimension- one abelian ideal, admitting left-invariant SKT metrics has been recently obtained in [5].
In [27] the authors studied Hermitian metrics whose Levi-Civita and Chern connection have curvature tensors satisfying all the symmetry conditions of a Kähler metric. Hermitian metrics with the Bismut connection being “ Kähler-like”, namely, satisfying the first Bianchi identity and the condition \(R^B(x,y,z,w)=R^B(Jx,Jy,z,w)\), for every tangent vectors x, y, z, w, have been studied, in [4], investigating this property on 6-dimensional solvmanifolds with holomorphically trivial canonical bundle.
In [30] Zhao and Zheng show that if the curvature tensor of the Bismut connection satisfies the symmetry conditions
for any tangent vectors x, y, z, w in X, then the Hermitian metric must be SKT. In [28] a classification for compact non-Kähler Hermitian manifolds whose Bismut connection is Kähler-like in complex dimension 3 and those with degenerate torsion in higher dimensions is given.
An evolution equation of SKT metrics is given by the pluriclosed flow, introduced by Streets and Tian in [19]. In recent years the pluriclosed flow has been an active subject of study, and already many regularity and convergence results have been proved in [20, 21]. A natural question is to see if the Bismut Kähler-like condition is preserved by the flow.
In this paper we study Hermitian metrics whose Bismut connection \(\nabla ^B\) satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bismut connection. In particular, as a consequence of Theorems 3.1 and 3.2 we show that, if X is a complex manifold and g is a Hermitian metric such that the Bismut connection satisfies the first Bianchi identity, then,
Moreover, we show in Proposition 3.11 that the existence of these metrics is not open under small deformations of the complex structure.
Specializing to complex dimension 2 we are able to characterize these metrics as follows
Theorem A
Let X be a complex surface and g be a Hermitian metric. Then, g is Vaisman if and only if g is a SKT metric and the Bismut connection satisfies the first Bianchi identity.
Where we recall that a Vaisman metric \(\omega \) on a complex manifold X is a Hermitian metric satisfying \(d\omega =\theta \wedge \omega \) for some d-closed 1-form \(\theta \) with \(\nabla ^{LC}\theta =0\).
In [6] a generalization of Vaisman metrics, called metrics with Lee potential, is introduced, and in Sect. 3.1 we study this condition in relation with the SKT condition.
In Sect. 4 we construct new examples of Hermitian manifolds satisfying the Bismut Kähler-like condition, using the characterization of SKT simply connected almost abelian Lie groups. In order to do this we compute explicitly the components of the Bismut connection. Moreover, we give conditions on the structure equations of almost abelian Lie groups in order to have Kähler and flat Hermitian metrics.
In the last Section we study the Bismut Kähler-like condition in relation to the pluriclosed flow, discussing its behavior on complex dimension 2 and on almost abelian Lie groups. In particular, we prove the following
Theorem B
Let X be a compact complex surface admitting a Vaisman metric \(\omega _0\) with constant scalar curvature, then the pluriclosed flow starting with \(\omega _0\) preserves the Vaisman condition.
2 Preliminaries
Let (X, J) be a complex manifold of complex dimension n and let g be a Hermitian metric on X with associated fundamental form \(\omega (\cdot \,,\cdot )=g(\cdot ,J\,\cdot )\) . An affine connection is called Hermitian if it preserves the metric g and the complex structure J. In particular, Gauduchon in [14] proved that there exists an affine line \(\left\{ \nabla ^t\right\} _{t\in {\mathbb {R}}}\) of canonical Hermitian connections, passing through the Chern connection and the Bismut connection; these connections are completely determined by their torsion. Let \(\nabla \) be a Hermitian connection and \(T(x,y)=\nabla _xy-\nabla _yx-[x,y]\) be its torsion, we denote with the same symbol
Then the Chern connection \(\nabla ^{Ch}\) is the unique Hermitian connection whose torsion has trivial (1, 1)-component and the Bismut connection (also called Strominger connection) \(\nabla ^B\) is the unique Hermitian connection with totally skew-symmetric torsion. In particular, the torsion of the Bismut connection satisfies
where \(d^c=-J^{-1}dJ\).
A Hermitian metric \(\omega \) is called strong Kähler with torsion (SKT for brevity) or pluriclosed if \(T^B\) is a closed 3-form, namely \(dT^B=0\), or equivalently \(dd^c\omega =0\).
Recall that the trace of the torsion of the Chern connection is equal to the Lee form of \(\omega \) (cf. [13]), that is the 1-form defined by
where \(d^*\) is the adjoint of the exterior derivative d with respect to \(\omega \), or equivalently \(\theta \) is the unique 1-form satisfying
A Hermitian metric \(\omega \) is called Gauduchon if \(dd^c\omega ^{n-1}=0\), or equivalently \(d^*\theta =0\). In particular, in dimension 2 Gauduchon and SKT metrics coincide. We recall the following
Definition 2.1
A Hermitian metric \(\omega \) on X is called locally conformally Kähler (lck for brevity) if
where \(\alpha \) is a d-closed 1-form. In particular, \(\alpha =\frac{1}{n-1}\theta \) and \(\theta \) is d-closed.
A locally conformally Kähler metric is called Vaisman if the Lee form is parallel with respect to the Levi-Civita connection \(\nabla ^{LC}\), namely
In particular, it is immediate to see that Vaisman metrics are Gauduchon and the norm of the Lee form \(|\theta |\) with respect to \(\omega \) is constant.
The Chern and Bismut connections are related to the Levi-Civita connection \(\nabla ^{LC}\) by
If we denote with
the curvature tensor of type (1, 3) and (0, 4), respectively, of a connection \(\nabla \) then we have the following identities involving the torsion and the curvature of the Bismut connection (cf. [15]) which will be useful in the following (cf. [15, Formulas (3.20), (3.21)])
In particular, this shows that the Bismut connection does not satisfy the first Bianchi identity in general. We recall the following definition (cf. [4, Definition 4], [27, 30])
Definition 2.2
The Bismut connection \(\nabla ^B\) is called Kähler-like if it satisfies the first Bianchi identity
and the type condition
In [4] the authors study these two conditions for the canonical connections considered by Gauduchon on 6-dimensional solvmanifolds with invariant complex structures, trivial canonical bundle and invariant Hermitian metrics.
In [30] Zhao and Zheng show that if the curvature tensor of the Bismut connection satisfies the symmetry conditions
for any tangent vectors x, y, z, w in X, then the Hermitian metric must be SKT. We will show that the previous condition for the curvature of the Bismut connection to be symmetric when the first and the third position are interchanged is stronger than the first Bianchi identity. In fact, in a private communication Zhao and Zheng showed that those two conditions are equivalent to the vanishing of the curvature \(R^B\).
3 Bismut Käher-like condition
In this section we investigate the properties (4) and (5) in the definition of Bismut Kähler like Hermitian metrics. In particular, we focus on the relations between the first Bianchi identity for the Bismut connection, the SKT condition and the parallelism of the torsion of the Bismut connection, specializing then our considerations to dimension 2, giving a characterization for Vaisman metrics.
Theorem 3.1
Let X be a complex manifold with a SKT Hermitian metric g such that the Bismut connection satisfies the first Bianchi identity. Then,
Proof
By hypothesis, \(dT^B=0\) and the first Bianchi-identity holds, hence by Formula (3) one has that for any tangent vectors x, y, z, u,
and by Formula (2)
Hence, one gets
Now notice that both sides of the equality are tensorial and on the left hand side the expression is symmetric in x, y, z while on the right hand side is antisymmetric in x, y, z. Therefore in order to be equal they must vanish. Hence, \(\nabla ^BT^B=0\). \(\square \)
Theorem 3.2
Let X be a complex manifold and let g be a Hermitian metric such that \(\nabla ^BT^B=0\). Then, the Bismut connection satisfies the first Bianchi identity if and only if g is SKT.
Proof
If \(\nabla ^BT^B=0\) then by Formula (2)
and so by Formula (3)
Therefore, the Bismut connection satisfies the first Bianchi identity if and only if g is SKT. \(\square \)
Hence, by putting together Theorems 3.1 and 3.2 we obtain the following
Corollary 3.3
Let X be a complex manifold and let g be a Hermitian metric such that the Bismut connection satisfies the first Bianchi identity. Then,
Now we consider some relations with respect to the Levi–Civita connection
Theorem 3.4
Let X be a complex manifold and g a Hermitian metric. If the Bismut connection satisfies the first Bianchi identity and g is SKT then
Proof
If the Bismut connection satisfies the first Bianchi identity and g is SKT (\(dT^B=0\)), then by Theorem 3.1 we have \(\nabla ^BT^B=0\) and so by Formula (3)
then by [15, Formula (3.18)]
\(\square \)
Notice that if the dimension of X is 2, then (cf. [15, (2.14)])
hence, on a complex surface
Then as a consequence of Theorem 3.4 one has
Corollary 3.5
Let X be a complex surface and g be a Gauduchon metric such that the Bismut connection satisfies the first Bianchi identity. Then, g is Vaisman.
In fact, we can prove that also its converse is true and show Theorem A.
Proof of Theorem A
Since Vaisman metrics are Gauduchon (or equivalently SKT in complex dimension 2), we just need to prove that if g is Vaisman then the Bismut connection satisfies the first Bianchi identity. Recall that on complex surfaces by [1, Appendix A]
Since, by hypothesis \(\nabla ^{LC}\theta =0\) then \(\nabla ^BT^B=\nabla ^{LC}T^B=0\), hence by Theorem 3.2 the Bismut connection satisfies the first Bianchi identity. \(\square \)
Notice that, up to now we have not used the second part of the definition of Bismut Kähler-like metrics. First of all, notice that if
then for the Ricci form of the Bismut connection we have
where \(\left\{ e_i\right\} \) denotes an orthonormal basis of the tangent space. Namely, \(\rho ^B\) is a (1, 1)-form. In particular,
where \((\beta )^{1,1}\) denotes the (1, 1) component of a 2-form \(\beta \). Hence, since
where \(\rho ^{Ch}\) is the Ricci form of the Chern connection, one has that
As a consequence,
Proposition 3.6
Let X be a complex surface and let g be a Hermitian metric such that
for every tangent vectors x, y, z, w. Then, g is locally conformally Kähler.
Proof
Let \(\omega \) be the fundamental form associated to g. Then, \(d\omega =\theta \wedge \omega \) and so we need to prove that \(d\theta =0\).
First of all, notice that in any dimension \(d\theta \) is a primitive 2-form, indeed differentiating \(d\omega ^{n-1}=\theta \wedge \omega ^{n-1}\) one has
where \(L=\omega \wedge -\) is the operator of multiplication by \(\omega \) acting on forms. Therefore, we compute
where in the second equality we have used for instance [25, Formula (A8)]. Then,
\(\square \)
3.1 SKT metrics with Lee potential
In [6] a generalization of lck metrics with potential (and so also of Vaisman metrics) is introduced and it is shown that these metrics exist on Calabi-Eckmann manifolds.
Definition 3.7
[6, 28] A Hermitian manifold (X, J, g) is called LP (or equivalently it has Lee potential) if the (1, 0)-part of the Lee form \(\theta \) of g, namely \(\eta := \theta ^{1,0}\), satisfies
for some non-zero constant c.
If in addition \(\nabla ^BT^B=0\) the metric g is called Generalized Calabi-Eckmann (GCE for short).
As a consequence of Theorem 3.2 the Bismut connection of a Generalized Calabi-Eckmann SKT metric satisfies the first Bianchi identity.
We show that if a SKT metric is LP then all its powers are \(\partial \overline{\partial }\)-closed.
Proposition 3.8
Let X be a complex manifold of complex dimension n endowed with a SKT metric \(\omega \) with Lee potential. Then,
In particular, \(\omega \) is \(k^{\text {th}}\)-Gauduchon for every \(1\le k\le n-1\) i.e.,
and astheno-Kähler, i.e., \(\partial \overline{\partial }\omega ^{n-2}=0\).
Proof
First of all, notice that for any k
Since, \(\omega \) is SKT and LP one has that
hence
Moreover, by the SKT and LP conditions one has also
concluding the proof. \(\square \)
Remark 3.9
Observe that in [28, Remark 2] it was proved that a Hermitian metric that is both SKT and Gauduchon then it is \(k^{\text {th}}\)-Gauduchon.
The LP assumption is fundamental in the previous Proposition, indeed in the following example we exhibit a manifold with a SKT metric \(\omega \) which is not LP and \(\partial \overline{\partial }\omega ^{n-2}\ne 0\), namely it is not astheno-Kähler.
Example 3.10
Let (G, J) be the 4-dimensional nilpotent Lie group equipped with a left invariant complex structure J with structure equations
with respect to a left invariant unitary coframe \(\left\{ \varphi ^i\right\} _{i=1,\ldots ,4}\), where \(\lambda _1,\lambda _2, a\) are real numbers and \(\lambda _1,\lambda _2>0\). It is easy to see that the left-invariant Hermitian metric \( \omega :=\frac{i}{2}\sum _{j=1}^4\varphi ^j\wedge {\bar{\varphi }}^j\) is SKT. Now we show that it is not LP.
By explicit computation one can show that the Gauduchon torsion (1, 0)-form \(\eta \) is
In particular, computing
and
One gets that \(\partial \omega =c\eta \wedge \partial {\bar{\eta }}\) for some non-zero constant c if and only if \(\lambda _1=-ia=0\) but \(\lambda _1>0\) so the metric \(\omega \) is not LP.
Moreover, notice that the metric \(\omega \) is not astheno-Kähler, indeed one has
3.2 Behavior under small deformations
In this section we partially answer to a question proposed in [4] about the stability of the Bismut Kähler-like property under small deformations of the complex structure. In particular, we show the following
Proposition 3.11
The existence of a SKT metric with Bismut connection satisfying the first Bianchi identity and with \(\nabla ^BT^B=0\) on compact complex manifolds is not an open property under small deformations of the complex structure.
Proof
Let \(X={\mathbb {S}}^3\times {\mathbb {S}}^3\) and \({\mathbb {S}}^3\simeq \hbox {SU}(2)\) be the Lie group of special unitary \(2\times 2\) matrices and denote by \(\mathfrak {su}(2)\) its Lie algebra. Denote by \(\{e_1, e_2, e_3\}\), \(\{f_1, f_2, f_3\}\) a basis of the first copy of \(\mathfrak {su}(2)\), respectively of the second copy of \(\mathfrak {su}(2)\) and by \(\{e^1, e^2, e^3\}\), \(\{f^1, f^2, f^3\}\) the corresponding dual co-frames. Then we have the following commutation relations:
and the corresponding Cartan structure equations
Define a complex structure J on X by setting
Therefore a complex co-frame of (1, 0)-forms for J is given by
In particular the complex structure equations are given by
Note that (X, J) is a central Calabi-Eckmann threefold and in [26] it is showed that this complex manifold admits a Bismut-flat Hermitian metric, which in particular is SKT and satisfies the first Bianchi identity.
Now let \(J_t\) be the almost complex structure on X considered in [22] defined as
then, using the structure equations (8), a straightforward computation yields to
and consequently \(J_t\) is integrable. Set \(X_t=(X,J_t)\), in [22, Remark 3.6] it was proven that when \(|t|^2+\text {Re}\,t-\text {Im}\,t\ne 0\) then \(X_t\) does not admit any SKT metric. Therefore by Corollary 3.3, for such values of t, \(X_t\) does not admit any Hermitian metric whose Bismut connection satisfies the first Bianchi identity and with parallel torsion. \(\square \)
4 Bismut Kähler-like almost abelian Lie groups
In this Section, we study the existence of Hermitian metrics with Kähler-like Bismut connection on simply-connected, almost abelian Lie groups in order to give new examples besides the ones provided in [4] on 6-dimensional solvmanifolds with invariant complex structures, trivial canonical bundle and invariant Hermitian metrics. Let G be a simply-connected, almost abelian Lie group namely, its Lie algebra \({\mathfrak {g}}\) has a codimension-one abelian ideal \({\mathfrak {n}}\). In particular, notice that such a G is solvable. Let (J, g) be a left-invariant Hermitian structure on G, therefore there exists a basis \(\left\{ e_1,\,\ldots \,,e_{2n}\right\} \) on \({\mathfrak {g}}\) such that, setting \({\mathfrak {n}}_1=\text {Span}_{{\mathbb {R}}}\left\langle e_2,\,\ldots \,,e_{2n-1}\right\rangle \), one has
By [16] the complex structure J is integrable if and only if ad \(e_{2n}\) leaves \({\mathfrak {n}}_1\) invariant, and \(A := (\text {ad}\,e_{2n} )|_{{\mathfrak {n}}_1}\) commutes with \(J_1:=J|_{{\mathfrak {n}}_1}\) . Hence one has
with \(a\in {\mathbb {R}}\), \(v\in {\mathfrak {n}}_1\), \(A\in \mathfrak {gl}({\mathfrak {n}}_1)\) and \([A,J_1]=0\).
In particular, we have the following non-trivial Lie brackets
for any \(x\in {\mathfrak {n}}_1\).
We fix a real inner product g on \({\mathfrak {g}}\) with an orthogonal decomposition
and
and a compatible integrable complex structure J such that \(Je_1=e_{2n}\) and \(J({\mathfrak {n}}_1)\subset {\mathfrak {n}}_1\).
We recall that by, [5, Lemma 4.2] the metric g is SKT if and only if
In order to study the existence of Bismut Kähler-like metrics we need the expression of the Bismut connection. By using the formula (see [10])
we get the following
Lemma 4.1
Let G be an almost abelian Lie group. Then, with the previous notations, the only non-zero components of the Bismut connection are
with \(x,y\in {\mathfrak {n}}_1\) and \(S(A):=\frac{1}{2}(A+A^t)\).
In particular if we assume that \(A\in \mathfrak {so}({\mathfrak {n}}_1)\) the non-trivial components reduce to
If \(y\in {\mathfrak {n}}_1\), by explicit computations one gets that that the only non-trivial components of the Bismut curvature tensor are
together with their symmetries. We can now prove
Theorem 4.2
Let G be an almost abelian Lie group and assume that \(A\in \mathfrak {so}({\mathfrak {n}}_1)\). Then, the Bismut connection is Kähler-like if and only if Av is g-orthogonal to \({\mathfrak {n}}_1\).
Proof
One can check that the first Bianchi identity holds if and only if \(g(v,Ay)=0\) for any \(y\in {\mathfrak {n}}_1\). Indeed, for instance
and similarly for the other relations. For the J-invariance in the first two components of \(R^B(\cdot ,\cdot ,\cdot ,\cdot )\) one has the same conclusion, for example
and so on. Hence, the Bismut connection is Kähler-like if and only if \(g(v,Ay)=0\) for every \(y\in {\mathfrak {n}}_1\) if and only if Av is g-orthogonal to \({\mathfrak {n}}_1\). \(\square \)
Remark 4.3
Notice that in general these Bismut Kähler-like metrics are not Kähler. More precisely, the torsion of the Bismut connection is given by
for \(y\in {\mathfrak {n}}_1\).
And so the only non-zero components of the torsion 3-form are
In particular, one can check explicitly that \(\nabla ^BT^B=0\). Hence, these exist explicit examples of metrics that are SKT, with \(\nabla ^BT^B=0\) but they do not satisfy the condition \(R^D(x,y,z,w) = R^D(z,y,x,w),\) for every tangent vectors x, y, z, w.
Remark 4.4
We notice that having \(A\in \mathfrak {so}({\mathfrak {n}}_1)\) is not necessary in order to have a Hermitian metric with Bismut Kähler-like connection on an almost abelian simply-connected Lie group. Indeed, by [4] the Lie algebra \({\mathfrak {h}}_8\) with structure equations (0, 0, 0, 0, 12) is an almost abelian Lie algebra with Bismut Kähler-like connection but the corresponding matrix A is not antisymmetric.
We now discuss the existence of a compact quotient on an explicit example in dimension 6. First of all notice that if \({\mathfrak {g}}\) is an almost abelian Lie algebra then it is unimodular (namely all the adjoint maps are traceless) if and only if \(a+\text {tr}\,A=0\).
If \(A \in \mathfrak {so}({\mathfrak {n}}_1)\), then \({\mathfrak {g}}\) is unimodular if and only if \(a=0\).
Example 4.5
Let \({\mathfrak {g}}\) be an almost abelian Lie algebra of dimension 6 with non trivial brackets
Hence we have
with \(v=e_2\), \(a=0\) and
We set \(\varphi (t)=e^{t{\tilde{A}}}\), by [9] \({\mathfrak {g}}\) admits a lattice if and only if there exists \(t_0\in {\mathbb {R}}\) such that \(\varphi (t_0)\) is conjugate to an integral matrix. Set \(t_0:=\pi \), then
This matrix is conjugate to
Hence, there exists a lattice \(\Gamma \) in G such that \(\Gamma \backslash G\) is an almost abelian solvmanifold of dimension 6 admitting a Bismut Kähler-like Hermitian metric.
In relation to the condition
for every tangent vector x, y, z, w. we can prove the following
Theorem 4.6
An almost abelian Lie group G admits a left-invariant SKT metric satisfying (9) if and only if
Proof
We have
Since
and for every \(x\in {\mathfrak {n}}_1\)
where \(S(A):=\frac{1}{2}(A+A^t)\), we have,
hence \(G^B(e_1,e_1,e_{2n},e_{2n})=0\) if and only if \(a=0\) and \(v=0\). Similarly, assuming that \(a=0\) and \(v=0\) we get, for every \(x\in {\mathfrak {n}}_1\)
hence \(G^B(e_1,e_1,x,x)=0\) for every \(x\in {\mathfrak {n}}_1\) if and only if \(g(S(A)Jx,e_j)=0\) for every \(x\in {\mathfrak {n}}_1\) and every \(j=1,\ldots ,2n\) (notice that for \(j=1\) and \(j=2n\) it is obvious) if and only if \(S(A)=0\).
Thus, if g satisfies (9) then we have just proven that \(a=0\), \(v=0\), and \(A\in \mathfrak {so}({\mathfrak {n}}_1)\). Now we show the viceversa, suppose that \(a=0\), \(v=0\) and \(A\in \mathfrak {so}({\mathfrak {n}}_1)\) then by similar computations one can show that the only non trivial components of the Bismut connection are, for every \(y\in {\mathfrak {n}}_1\)
hence, by definition
for every \(x,y,z\in {\mathfrak {g}}\), hence \(R^B(x,y,z,w)=0\) for every \(x,y,z,w\in {\mathfrak {g}}\) concluding the proof. \(\square \)
Notice that, in particular, we have proven that an almost abelian Lie group G has \(R^B(x,y)z=0\) for every \(x,y,z\in {\mathfrak {g}}\) if and only if
in particular almost abelian Lie groups satisfying (9) are Bismut flat. In fact, we prove the following
Theorem 4.7
An almost abelian Lie group G with
is Kähler. In particular, the only almost abelian examples satisfying (9) are Kähler and flat.
Proof
If \(v=0\) and \(A\in \mathfrak {so}({\mathfrak {n}}_1)\) then the only non-zero commutators become
where \(x\in {\mathfrak {n}}_1\). Hence, computing explicitly the Bismut connection for (g, J) as before one gets that the only non-trivial components are
where \(y\in {\mathfrak {n}}_1\).
Computing explicitly the torsion
one gets \(T^B=0\). For instance we compute here, for \(y\in {\mathfrak {n}}_1\),
\(\square \)
5 Bismut-Kähler like condition and pluriclosed flow
Let (X, J) be a complex manifold of complex dimension n. In [19] the authors introduce a family of flows called Hermitian curvature flows, among these a particular one is the so called pluriclosed flow which is defined by the equation
where \(\left( \rho ^B(\omega )\right) ^{1,1}\) denotes the (1, 1)-part of the Ricci form of the Bismut connection and \(\omega _0\) is a fixed Hermitian metric. It is easy to see that this flow preserves the SKT condition.
In this section we study the behavior of the Bismut Kähler-like condition under the pluriclosed flow on complex surfaces and on almost abelian Lie groups considered in the previous section.
5.1 Pluriclosed flow on Vaisman surfaces
We recall that a Vaisman metric on a complex manifold \((X\,,J)\) is a Hermitian metric \(\omega \) such that
where \(\theta \) is a 1-form.
In fact, a Vaisman structure on a complex manifold is uniquely determined (up to a positive constant) by its Lee form \(\theta \) via the following
On complex dimension 2, Belgun in [7] classified those compact complex surfaces admitting a Vaisman metric, and they are properly elliptic surfaces, primary and secondary Kodaira surfaces, elliptic Hopf surfaces and Hopf surfaces of class 1.
Notice that in particular, on a compact complex surface, Vaisman metrics are SKT. We ask whether the Vaisman condition is preserved along the pluriclosed flow. We answer this question in the case the initial metric has constant scalar curvature.
In order to prove Theorem B we first need the following Lemma.
Lemma 5.1
Let X be a compact complex surface and let \(\omega \) be a Vaisman metric on X with Lee form \(\theta \). Then, the Ricci form of the Chern connection is
for some \(h\in {\mathcal {C}}^\infty (X,{\mathbb {R}})\).
Moreover, the scalar curvature of \(\omega \) is constant if and only if h is constant and, in particular, in such a case \(c_1(X)=0\).
Proof of Theorem B
Let \((\omega _0,\theta _0)\) be a Vaisman structure on X with constant scalar curvature; then, by Lemma 5.1, we have
Moreover, since \(\omega _0\) is Vaisman, then
where with \(|\cdot |_0\) we denote the norm with respect to \(\omega _0\) and in particular \(|\theta _0|_0\) is constant.
We claim that the metrics
for a suitable positive function f(t) depending only on t with \(f(0)=1\), are Vaisman and they are solutions of the pluriclosed flow.
First of all notice that the corresponding Lee form of \(\omega _t\) is
indeed, \(d\omega _t=\theta _t\wedge \omega _t\) and clearly \(\theta _t\) is closed, hence they are locally conformally Kähler.
Moreover, the Lee field \(\theta ^{\sharp _t}_t\) with respect to \(\omega _t\) is
hence by [18, Theorem 1] the metrics \(\omega _t\) are Vaisman.
In fact, recall that on a compact complex manifold admitting Vaisman metrics, the Lee vector fields of all Vaisman structures are holomorphic, and coincide up to a positive multiplicative constant (cf. [23] and [17]).
Moreover, by a straightforward computation,
therefore, the Ricci forms of the Chern connection of \(\omega _t\) and \(\omega _0\) must coincide
Hence, the Ricci forms of the Bismut connection of \(\omega _t\) and \(\omega _0\) are related by
Notice that, since \(\omega _t\) is lck then \(\rho ^{B}_{\omega _t}\) is a closed (1, 1)-form.
For the pluriclosed flow we have
hence it is necessary to have \(c_1(X)=0\) to have \(\omega _t\) solving
because we have
so \(\rho ^{Ch}_{\omega _0}\) is an exact form and, up to a constant, it represents the first Chern class of X.
Now, by Lemma 5.1
for some constant h. Hence,
and so
The equation of the pluriclosed flow reduces to find a solution f(t) of
or equivalently
In fact, the equation
admits a unique solution \(f(t)>0\) and with \(f(0)=1\). Therefore, \((\omega _t,\theta _t)\) are Vaisman metrics and solutions of the pluriclosed flow. \(\square \)
Remark 5.2
We notice that, if X is a compact complex surface admitting a Vaisman metric \(\omega _0\) with constant scalar curvature, then the pluriclosed flow starting with \(\omega _0\) preserves such a condition. Indeed, by Lemma 5.1 one has for every \(t\ne 0\)
for some smooth function \(h_t\) on X, and for \(t=0\)
with \(h_0\) constant by hypothesis.
Then, by the proof of Theorem B we have that
therefore \(h_t=f(t)h_0\) for every t, giving \(h_t\) constant and so \(\omega _t\) has constant scalar curvature.
We now give a proof of Lemma 5.1
Proof of Lemma 5.1
Let \(\omega \) be a Vaisman metric on X. Then, by [2, Lemma 4.4] the Ricci forms of the Bismut and Weyl connections coincide
where the Weyl connection determined by the Hermitian structure g of X is the unique torsion-free connection \(\nabla ^W\) such that \(\nabla ^Wg=\theta \otimes g\). In fact, the Weyl connection is related with the Levi-Civita connection of g via
Since \(\rho ^{Ch}=\rho ^B+dJ\theta =\rho ^W+dJ\theta \) we just need to prove that
for some smooth function h on X.
Since the metric is Vaisman then the Ricci tensor \(\text {Ric}^W\) is symmetric and one has (cf. [3])
where \(\text {Ric}^{LC}\) is the Ricci tensor of the Levi-Civita connection.
Therefore, for the Ricci form of the Weyl connection we can compute
Since \(\theta \) is parallel with respect to the Levi-Civita connection we have that \(R^{LC}(x,y,\theta ^\sharp )=0\) for any tangent vectors x, y. Hence, \(\text {Ric}^{LC}(\theta ^\sharp ,\theta ^\sharp )=0\) and so
Now assume, without loss of generality, that \(|\theta |=1\) and take a orthonormal basis \(\theta ,J\theta ,\xi ,J\xi \). Then, we have
and
for some smooth functions h, k. Since, \(\rho ^W(\theta ^\sharp ,J\theta ^\sharp )=0\), then \(k=0\) and so
Now, notice that the scalar curvature s of g is constant if and only if the same holds for the trace b of \(\rho ^B\). This follows by [2, Formula (2.12)],
having \(|d\omega |^2=|\theta |^2\) constant for a Vaisman metric on a complex surface. Therefore, b is constant if and only if the same holds for the function h, indeed
where we have used that \(\rho ^B=\rho ^W=h\xi \wedge J\xi \). \(\square \)
5.2 Pluriclosed Flow on almost abelian Lie groups
In [5] it is shown that the pluriclosed flow on almost abelian Lie algebras reduces to
where
and
We prove the following
Theorem 5.3
Let G be a simply-connected almost abelian Lie group. If \(A_0 \in \mathfrak {so}({\mathfrak {n}}_1)\), then the pluriclosed flow preserves the Bismut Kähler-like condition.
Proof
Notice that if \(A_0 \in \mathfrak {so}({\mathfrak {n}}_1)\) then by uniqueness of the solution \(A(t)\in \mathfrak {so}({\mathfrak {n}}_1)\). Hence, if we assume \(A_0 \in \mathfrak {so}({\mathfrak {n}}_1)\) one gets
and
We show that if \(A_0 \in \mathfrak {so}({\mathfrak {n}}_1)\) and at \(t=0\) the metric is Bismut Kähler-like then it remains Bismut Kähler-like along the flow. Indeed, if if \(A_0 \in \mathfrak {so}({\mathfrak {n}}_1)\) and at \(t=0\) the metric is Bismut Kähler-like one has that \(A_0v_0\) is orthogonal to \({\mathfrak {n}}_1\) and by [5] \({\mathfrak {n}}_1\) is preserved along the flow. We compute, for any \(y\in {\mathfrak {n}}_1\)
hence
Then, if \(A_0v_0\) is orthogonal to \({\mathfrak {n}}_1\) by uniqueness of the solution it remains orthogonal along the flow and so the Bismut Kähler-like condition is preserved. \(\square \)
References
Agricola, I., Ferreira, A.C.: Einstein manifolds with skew torsion. Q. J. Math. 65(3), 717–741 (2014)
Alexandrov, B.: S: Ivanov, Vanishing theorems on Hermitian manifolds. Differ. Geom. Appl. 14(3), 251–265 (2001)
Alexandrov, B., Ivanov, S.: Weyl structures with positive Ricci tensor. Differ. Geom. Appl. 18(3), 343–350 (2003)
Angella, D., Otal, A., Ugarte, L., Villacampa, R.: On Gauduchon connections with Kähler-like curvature. Commun. Anal. Geom (2018) (to appear). arXiv:1809.02632 [math.DG]
Arroyo, R.M., Lafuente, R.: The long-time behavior of the homogeneous pluriclosed flow. Proc. Lond. Math. Soc. 3(119), 266–289 (2019)
Belgun, F.A.: On the metric structure of some non-Kähler complex threefolds (2012). arXiv:1208.4021
Belgun, F.A.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317, 1–40 (2000)
Bismut, J.-M.: A local index theorem for non-Kähler manifolds. Math. Ann. 284(4), 681–699 (1989)
Bock, C.: On low-dimensional solvmanifolds. Asian J. Math 20(2), 199–262 (2016)
Dotti, I., Fino, A.: HyperKähler torsion structures invariant by nilpotent Lie groups. Class. Quant. Grav. 19(3), 551–562 (2002)
Fino, A., Parton, M., Salamon, S.: Families of strong KT structures in six dimensions. Comment. Math. Helv. 79(2), 317–340 (2004)
Fino, A., Otal, A., Ugarte, L.: Six-dimensional solvmanifolds with holomorphically trivial canonical bundle. Int. Math. Res. Not. IMRN 24, 13757–13799 (2015)
Gauduchon, P.: La \(1\)-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267(4), 495–518 (1984)
Gauduchon, P.: Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B (7) 11(2), 257–288 (1997)
Ivanov, S., Papadopoulos, G.: Vanishing theorems and string backgrounds. Classi. Quant. Grav. 18(6), 1089–1110 (2001)
Lauret, J., Valencia, E.A.R.: On the Chern-Ricci flow and its solitons for Lie groups. Math. Nachr. 288, 1512–1526 (2015)
Madani, F., Moroianu, A., Pilca, M.: LcK structures with holomorphic Lee vector field on Vaisman-type manifolds, arXiv:1905.07300 [math.DG], (2019)
Moroianu, A., Moroianu, S., Ornea, L.: Locally conformally Kähler manifolds with holomorphic Lee field. Differ. Geom. Appl. 60, 33–38 (2018)
Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 16, 3101–3133 (2010)
Streets, J., Tian, G.: Regularity theory for pluriclosed flow. C. R. Math. Acad. Sci. Paris 349(1–2), 1–4 (2011)
Streets, J., Tian, G.: Regularity results for pluriclosed flow. Geom. Topol. 17(4), 2389–2429 (2013)
Tardini, N., Tomassini, A.: On geometric Bott-Chern formality and deformations. Ann. Mat. Pura Appl. (4) 196(1), 349–362 (2017)
Tsukada, K.: Holomorphic forms and holomorphic vector fields on compact generalized Hopf manifolds. Compositio Mathematica 93(1), 1–22 (1994)
Ugarte, L.: Hermitian structures on six-dimensional nilmanifolds. Transform. Groups 12(1), 175–202 (2007)
Vaisman, I.: On some variational problems for \(2\)-dimensional Hermitian metrics. Ann. Glob. Anal. Geom. 8(2), 137–145 (1990)
Wang, Q., Yang, B., Zheng, F.: On Bismut flat manifolds (2016). arXiv:1603.07058 [math.DG]
Yang, B., Zheng, F.: On curvature tensors of Hermitian manifolds. arXiv:1602.01189 (to appear in Comm. Anal. Geom)
Yau, S.T., Zhao, Q., Zheng, F.: On Strominger Kähler-like manifolds with degenerate torsion (2019). arXiv:1908.05322 [math.DG]
Zhao, Q., Zheng, F.: Complex nilmanifolds and Kähler-like connections (2019). arXiv:1904.09707 [math.DG]
Zhao, Q., Zheng, F.: Strominger connection and pluriclosed metrics (2019). arXiv:1904.06604 [math.DG]
Acknowledgements
The authors would like to thank Luigi Vezzoni and Mihaela Pilca for useful discussions and suggestions. The authors would like to thank also Quanting Zhao and Fangyang Zheng for useful comments. The authors are grateful to Jeffrey Streets for pointing out an inaccuracy in the first version of Theorem B. The authors are supported by Project PRIN 2017 “Real and complex manifolds: Topology, Geometry and Holomorphic Dynamics”, by project SIR 2014 AnHyC “Analytic aspects in complex and hypercomplex geometry” code RBSI14DYEB, and by GNSAGA of INdAM.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fino, A., Tardini, N. Some remarks on Hermitian manifolds satisfying Kähler-like conditions. Math. Z. 298, 49–68 (2021). https://doi.org/10.1007/s00209-020-02598-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-020-02598-2