DEFORMATIONS OF STRONG KÄHLER WITH TORSION METRICS

Existence of strong Kähler with torsion (SKT) metrics on complex manifolds has been shown to be unstable under deformations. We develop a method to compute the complex exterior differentials along a curve of complex manifolds, originating from a base complex manifold, in function of the complex exterior differentials defined on the base complex manifold. We apply such a method in order to find a necessary condition to the existence of SKT metrics on the curve of complex manifolds.


Introduction
In this paper, we make use of classical deformation theory and curves of complex structures to prove stability conditions for SKT metrics once the base complex manifold undergoes a deformation of the complex structure. By results of Kodaira and Spencer, we know that the Kähler condition of a manifold, i.e. admitting a metric with closed fundamental form, is stable under infinitesimal deformations of the complex structure. Therefore, it is straightforward to consider those notions that generalize the Kähler condition which naturally arise in the Hermitian setting and study their stability under deformations.
Let (M, J, g) be a Hermitian manifold of complex dimension n. Depending on the closedness of the fundamental form ω (or its powers) of g with respect to certain differential operators, specific structures arise. If ∂∂ω = 0, the metric is said to be strong Kähler with torsion (SKT). Another notion which generalizes Kählerness is the balanced condition, i.e., dω n−1 = 0, or equivalently, being ω real, ∂ω n−1 = 0. It is in fact a particular case of a p-Kähler metric, i.e., dω p = 0, for 2 ≤ p ≤ n − 1. In respectively [5] and [1], it is proved that the existence of SKT and p-Kähler metrics is not stable, once the base complex manifold is deformed via a smooth family of complex structures.
Since the existence of SKT metrics on complex manifolds is not stable under deformations, it is worth investigating under which circumstances a SKT metric exists on a deformed complex manifold. In this paper, we show a necessary condition to the existence of SKT metrics on a curve of complex manifolds, see Theorem 5.1. To prove our result, we develop a method to compute the complex differentials ∂ t and ∂ t acting on functions along a curve of complex manifolds (M, J t ), in function of the complex differentials ∂ 0 and ∂ 0 on the base complex manifold (M, J 0 ), and in function of the (0, 1)-differential form with values in the holomorphic tangent bundle which describes the deformation of the complex structure. In Proposition 4.4 we write explicit formulas for computing ∂ t and ∂ t , and in Theorem 4.6 we approximate such formulas in a way to be applied to our main result. Note that it is not necessary to have any information of the complex coordinates of the deformed complex manifold to apply our method of computing ∂ t and ∂ t , neither it relies on using special algebraic structures, such as structure equations on Lie groups, to compute the complex differentials.
We remark that starting from Theorem 4.6, one could study the existence of other special metrics on complex manifolds, building different necessary conditions to the existence of such metrics.
Finally, we describe two example of nilmanifolds: the quotient of H(3; R) × H(3; R) by H(3; Z) × H(3; Z) and the Iwasawa manifold, i.e., the quotient of H(3; C) by H(3; Z[i]). The general theory that can be applied in these examples (see [4], [5], [2]) describes well the problem of the existence of a SKT metric on deformations of the base complex manifolds. We applied our main result to both the above examples, finding a nice setting in which this kind of computations can be carried on.
The paper is organized in the following way. In section 2, we fix the notation and recall some basic facts of complex geometry. Section 3 is a brief review of the classical deformation theory on complex manifolds, stating the main results of Kodaira and Kuranishi. In section 4, we recall some fundamental properties of curves of complex structures and develop our method of computing the complex differentials along the curve of complex manifolds. Section 5 is dedicated to the proof of our main result, i.e., a necessary condition to the existence of a SKT metric on a curve of complex manifolds. Finally, in section 6, we recall some cohomological properties of nilmanifolds and apply our main result to two explicit examples of nilmanifolds.

Notations and preliminaries
Let (M, J, g) be an Hermitian manifold, with J ∈ End(T M ) the integrable almost-complex structure on M and g a Riemannian metric on M compatible with J. Let ω be the (1, 1)−fundamental form associated to g given by ω(⋅, ⋅) = g(J(⋅), ⋅).
The metric g is said to be strong Kähler with torsion (or SKT) if where d = ∂ + ∂ is the decomposition induced by the complex structure.
Let π∶ E → M be a complex vector bundle of rank r over (M, J, g), a n-dimensional Hermitian manifold. For every p, q, let ⋀ p,q (E) ∶= ⋀ p,q (M ) ⊗ E be the bundle of the (p, q)-differential forms on M with values in E and let A p,q (E) ∶= Γ(M, ⋀ p,q (E)) be the space of its global C ∞ -sections.
If h is an Hermitian metric h on E, i.e. a smooth Hermitian scalar product on each fibre of E, let us identify h as a C-antilinear isomorphism between E and its dual E * and consider the usual C-antilinear Hodge * -operator on (M, J, g) with respect to g (see [6]). Then

Review of deformation theory of complex structures
We will recall the definitions both in the differentiable and complex settings. Let B be a domain of R m (resp. C m ) and {M t } t∈B a family of compact complex manifolds.
Definition 3.1. We say that M t depends differentiably (resp. holomorphically) on t ∈ B and that {M t } t∈B forms a differentiable (resp. holomorphic, or complex analytic) family if there is a differentiable (resp. complex) manifold M and a differentiable (resp. holomorphic) proper map π onto B such that (1) π −1 (t) = M t as a complex manifold for every t ∈ B, (2) the rank of the Jacobian of π is equal to the dimension (resp. complex dimension) of B at each point of M. More precisely, for each p 0 ∈ M there is a local diffeomorphism (resp. biholomorphism) from U j ∋ p 0 p ↦ (z 1 , . . . , z n , t 1 . . . , t m ) to a domain in C n × R m (resp. C n × C m ), where π(p) = (t 1 , . . . , t m ) are differentiable (resp. complex) coordinates around π(p), and such that for a fixed t ∈ R m (resp. C m ), (z 1 , . . . , z n ) are complex coordinates on M t . If U i ∩ U j ≠ ∅, the coordinates z i , z j are related via the transition functions f α ij , is differentiable (resp. holomorphic) in (z j , t) and holomorphic in z j for a fixed t. It follows from the definition that every M t , for t ∈ B, is a submanifold (resp. complex submanifold) of M.
Note that M and N are diffeomorphic as differentiable manifolds, see [10, pag. 147]. In the holomorphic setting, deformation theory proceeds then with the description of the tools which allow us to determine the existence and study in detail deformations of the complex structure of a complex manifold. Let M = {M t } t∈B , and π∶ M → B with B ∶= B r (0) ⊂ C m , be a holomorphic family over B. Let f α ij be the holomorphic transition functions on M.
If we denote by Θ t the sheaf of holomorphic vector fields on M t , we can define ∂Mt ∂t ν ∶= θ ij ν (t), with which is an element of the cohomology with values in Θ t , i.e. H 1 (M t , Θ t ). Let us set ∂ ∂t ∶= ∑ m j=1 ∂ ∂t ν . Definition 3.3. We define the infinitesimal deformation of M as Under assumptions on the cohomology space H 2 (M, Θ), Kodaira, Niremberger, and Spencer proved a theorem of existence of complex analytic deformations. Indeed, let B be the ball centered in 0 ∈ C of radius r > 0, i.e. B = B r (0) ⊂ C. Results of deformation theory assure that to each infinitesimal deformation ∂M ∂t t=0 ∈ H 1 (M, Θ) as in Definition 3.3, corresponds a unique (0, 1)-vector form on M which is ∂−closed. Indeed, let X be the sheaf of C ∞ vector fields on M , which can be thought as the (0, 0)−vector forms on M ; the Dolbeault isomorphism implies: We can actually describe the complex structure on each M t , t ∈ B via a C ∞ (0, 1)-vector form Ψ(t), defined starting from the local transition functions f α ij (see [10, pag. 150]). If such Ψ(t) is locally written as On the space of C ∞ vector forms a bracket can be defined in the following way. Let Ψ = ∑ ψ α ∂ α and Ξ = ∑ ξ α ∂ α be respectively (0, p)-and a (0, q)-vector forms, where ∂ α = ∂ ∂z α . Then In particular [ , ] is bilinear and satisfies the following if Ψ is a (0, p)-form, Ξ a (0, q)-form and Φ a (0, r)-form.
A classical results shows that the deformations of the complex structure on a compact complex manifold which give rise to integrable complex structures can be characterized according to the following theorem.
As by Theorem 3.4, existence of complex deformations of a compact complex manifold is assured if H 2 (M, Θ) = 0. However, if this property does not hold, a more general theory, known as Kuranishi theory, can be applied.
Let M be a compact complex manifold and define Fix an Hermitian metric h on M , extend it to A q and denote it by the same symbol h. Define and inner product on A q by where Ψ, Ξ ∈ A q , * is the C-antilinear Hodge operator. We also define the Laplacian on A q by where ∂ * is the adjoint operator of ∂ with respect to the Hermitian metric h. The space of harmonic forms is The Hodge theory induces a decomposition on the space A q as a direct sum of orthogonal subspaces: The operator G∶ A q → ◻A q is well defined and acts on A q as the projection onto ◻A q , whereas the operator H is the (well-defined) projection operator onto H q .
Theorem 3.6 (Kuranishi). Let M be a compact complex manifold, {η ν } a base for H 1 . Let Ψ(t) be the solution of the equation The space S is called the space of Kuranishi. Its defining property holds for a Ψ(t) satisfying (3.3) if and only Ψ(t) satisfies Maurer-Cartan equation (3.2). The proof of Theorem 3.6 shows that a (0, 1)-vector form Ψ(t) satisfying equation (3.3) can be constructed as a converging power series 3) assures that each term ψ µ can be computed as In general S can have singularities and hence may not have a structure of smooth manifold. Nonetheless, {M t } t∈S can be still be interpreted as a complex analytic family, see [7]. with L ∈ End(T M ) such that LJ + JL = 0 and det(I + L) ≠ 0 (see, e.g., [3]). It turns out thatĴ is integrable if and only if

Curves of Complex Structures
satisfies the Maurer-Cartan equation (3.2), i.e., Note that the curve t ↦ J t is a differentiable deformation of the complex structure J of M , according to Definition 3.1. Then, as recalled above, for − < t < , and some > 0, Finally, we set We need the following known lemma. We give the proof for completeness.
Lemma 4.1. Let z be a complex vector field of type (1, 0) and α be a complex 1-form of type (1, 0) with respect to the complex structure J. Then (I + L t )z and (I + L t )α (resp. (I + L t )z and (I + L t )α) are of type (1, 0) (resp. (0, 1)) with respect to the complex structure J t . Moreover, L t z and L t α are of type (0, 1) and L t z and L t α are of type (1, 0) with respect to the complex structure J.
Proof. Let us prove the lemma for z ∈ Γ(M, T 1,0 M ). The other cases are analogous. We get i.e., (I + L t )z is of type (1, 0) for the complex structure J t . Moreover, i.e., L t z is of type (0, 1) for the complex structure J.
Since det(I + L t ) ≠ 0 for − < t < , the map I + L t is an isomorphism between the spaces of complex vector fields of type (1, 0) (resp. (0, 1)) with respect to the complex structure J and complex vector fields of type (1, 0) (resp. (0, 1)) with respect to the complex structure J t . The same holds for the spaces of (1, 0) and (0, 1) forms. As another consequence of lemma 4.1, note that we have an explicit verifiction of Our goal is to understand how the differential operators ∂ t and ∂ t , namely the operators ∂ and ∂ with respect to the complex structure J t , act on functions.
Finally, since L t ∈ End(T M ), then L t z = L t z = Ψ t (z).
Analogously, we get the following. Then Proof. By making use of Lemma 4.1 and the characterization of J t (4.1), we compute Since L t z = Ψ t (z), we conclude. Now, fix a local frame {v 1 , . . . , v n } of complex vector fields of type (1, 0) and the corresponding dual local coframe {ξ 1 , . . . , ξ n } of (1, 0)-forms with respect to the complex structure J. Then, locally we have By the very definition of Ψ t and Lemma 4.1, we get that {v i + (ψ t ) j i v j } i is a local frame of complex vector fields of type (1, 0) and {ξ j +ξ i (ψ t ) j i } j is a local coframe of (1, 0)-forms with respect to the complex structure J t . Let us introduce the following matrix notation: v = (v 1 , . . . , v n ) T , ξ = (ξ 1 , . . . , ξ n ), ψ t = ((ψ t ) j i ) ij . The following products between matrices, vectors and covectors satisfy the rules of matrix products. Then, if f ∶ M → C is a C ∞ function, since ∂ t f is a (1, 0)-form with respect to J t , we have where a t = ((a t ) 1 , . . . , (a t ) n ) T is a vector of C ∞ local functions with complex values varying with t. Therefore and from Lemma 4.2, we get Since ψ(0) = 0, for t sufficiently small we get the following formula for ∂ t f . Analogously, we derive a formula for ∂ t f .
Proposition 4.4. Using the previous notation, we have: , Up to now, we did not really use the hypotesis of J t being a curve; the properties we showed for J t could be showed in the same way forĴ in a neighborhood of J. Since, in general, it is difficult to invert the matrices I + ψ t ψ t and I + ψ t ψ t , we need to simplify the previous formulas. We use the Taylor series from which we immediately derive the following Lemma.
Lemma 4.5. Using the previous notation, we have: Proof. The only thing to note is that (I + ψ t ψ t ) −1 = I + o(t).
Passing in local complex coordinates {z i } i , for v i = ∂ ∂z i and ξ i = dz i , we get the following general formulas for the differential operators ∂ t and ∂ t acting on functions. (4.5)

Main Result
Let (M, J, g, ω) be a Hermitian manifold and assume that the metric is SKT, i.e., ∂∂ω = 0. Let t ↦ J t be a smooth curve of complex structures on M such that J 0 = J. We are interested to find a necessary condition for the existence of a SKT metric g t on (M, J t ), which converges to g as t approaches 0. If ω t is a SKT metric on (M, J t ), i.e., ∂ t ∂ t ω t = 0, for t ∈ (− , ), then we obtain ∂ ∂t (∂ t ∂ t ω t ) t=0 = 0. Applying Theorem 4.6 to explicitly calculate in coordinates this necessary condition, we obtain the following theorem.
Theorem 5.1. Let (M, J, g, ω) be a Hermitian manifold of complex dimension n. Let t ↦ J t be a smooth curve of complex structures on M such that J 0 = J. Let Ψ t ∈ A 1 , locally written as Ψ t = (ψ t ) j i dz i ⊗ ∂ ∂z j , where {z i } i are local complex coordinates on (M, J), be associated to J t by equation (4.3). Locally, write (ψ t ) j i = tψ j i + o(t) and ω = ω ij dz i ∧ dz j . If ω t is a SKT Hermitian metric on (M, J t ) for t ∈ (− , ), > 0, ω 0 = ω and ω t is locally written as Proof. Since ∂ t ∂ t ω t = 0, for t ∈ (− , ), then we get ∂ ∂t (∂ t ∂ t ω t ) t=0 = 0. Let us compute explicitly ∂ ∂t (∂ t ∂ t ω t ) t=0 = 0, using the local complex coordinates {z i } i . First of all, let us set the notation Now, let us compute ∂ t ∂ t ω t , using equations (4.5). We begin calculating ∂ t ω t .
Then, we compute ∂ t ∂ t ω t . where If the metric ω t is SKT, then ∂ ∂t (∂ t ∂ t ω t ) t=0 = 0, and ∂ ∂t Thus, ∂ ∂t (∂ t ∂ t ω t ) t=0 = 0 if and only if the following three conditions holds:

Applications
The conditions in Theorem 5.1 hold in general for deformations on any SKT manifold. However, if we consider the class of nilmanifolds, exploiting the theory developed in [4] and [8], we can find a nice field of application for our results.
6.1. Deformations of Abelian complex structures on nilmanifolds. We recall some definitions. Let (M, J) is a nilmanifold with J Abelian complex structure and let m be the Lie algebra associated to M . As in [8] and [4], let us consider the sequence m * (0,k) ⊗ m 1,0 on which the linear operator ∂ acts in the following way. If V ∈ m 1,0 , U ∈ m 0,1 , we set so that ∂ ∶ m 1,0 → m * (0,1) ⊗ m 1,0 can be extented to a linear map on every space m * (0,k) ⊗ m 1,0 by It turns out that ∂ so defined coincides with the same opeartor defined in Section 1, in (2.1), on left invariants differentiable vector forms. Moreover, ∂ it is a differential, i.e ∂ k ○ ∂ k−1 ≡ 0. This allows one to define the jth cohomology of the complex {m * (0,•) ⊗ m 1,0 , ∂ } by The following result, due to [4], links (6.1) and the Dolbeault cohomology with values in the holomorphic tangent bundle. It is possible to find harmonic representatives for the Dolbeault cohomology with values in T 1,0 M by looking at the invariant cohomology. In fact, let us fix an appropriate invariant Hermitian metric g on M as in [4, §2]. Extending g to every m * (0,j) ⊗ m 1,0 we denote by ∂ * the formal adjoint of ∂ in the invariant setting with respect to g and by ∆ ∶= ∂∂ * +∂ * ∂ the Laplacian operator with respect to ∂. Let us set im ∂ j−1 as the orthogonal complement of im ∂ j−1 with respect to that fixed metric. Then, the following holds (see [8]). Theorem 6.3. The space im ∂ j−1 ⊂ ker ∂ j is a space of harmonic representatives for the Dolbeault cohomology H j (M, T 1,0 M ). Also, if µ ∈ m * (0,j) ⊗ m 1,0 , then ∂ * µ with respect to the L 2 -norm on the compact manifold M is equal to ∂ * µ with respect to the Hermitian inner product on the finite-dimensional vector space m * (0,j) ⊗ m 1,0 .
The bracket of vector forms (3.1) can be defined also in the invariant setting on cohomology classes in where by i V ′ dω we mean the contraction of the form dω with the vector field V ′ . We observe that this definition coincides with (3.1) on H 1 (M, T 1,0 M ).
We are now set to describe the Kuranishi method in this setting (as in [8]), which will yield a power series construction of the (0, 1)-vector form representing the deformations of the complex structure.

6.2.
Applications on examples. In the following example, we will apply the theory on Abelian nilmanifolds so far introduced together with Theorem 5.1 to study the SKT condition on products of the real Heisenberg group. so that (6.8) is a global coframe of (1, 0)-forms on M . The structure equations with respect to (6.8) are given by We notice that J is an Abelian complex structure (henceforth, integrable).
We will consider the following metric as the initial metric (6.10) ω = i 2 (ϕ 11 + ϕ 22 + ϕ 33 ) which locally can be written as From (6.9), it is clear that ∂∂ω = 0, i.e. ω is a SKT metric on M . By simple computations, we can find that the holomorphic coordinates which induce J on M are given by .
We can then rewrite locally (6.7) and the dual invariant vector fields {ϕ 1 , ϕ 2 , ϕ 3 }: 3 . We now proceed with making use of the tools we introduced in the beginning of this section. Since we can restrict ourselves to the invariant setting, we start by finding an invariant basis for H 1 .
It is a long but straightforward computation to substitute equations (6.30) in equation (6.31) and to apply Theorem 5.1 on ω t . One obtains that if ω t is SKT for t ∈ (− , )∖{0}, then ∂l ∂t t=0 = 0. Therefore, if ∂l ∂t t=0 ≠ 0, then ω t is not SKT for t ∈ (− , ) ∖ {0}. In fact, following the same method used in [5, Section 2] to prove the non-stability under small deformations of the SKT condition, one could proceed as follows. By direct computations using structure equations (6.29), one can show that M t does not admit a SKT invariant metric for t ≠ 0, since ∂ t ∂ t (ξ 3 t ∧ ξ 3 t ) = 0 iff t = 0. Again by [11,Proposition 21], if there exists a SKT metric on M t , then there is also a left invariant one. Thus, M t does not admit any SKT metric for t ≠ 0. Therefore, our method is weaker than the one in [5] in this case. Remark 6.8. As a conclusion, on a nilmanifold M with an invariant complex structure J 0 , an invariant coframe of (1, 0)-forms for the deformed complex structure J t yields affordable computations to verify if the manifold admits (invariant) SKT metrics, through the use of structure equations. In these cases, our necessary condition on the existence of SKT metrics gives us less information (due to the approximation of the first term of the Taylor expansion) and requires more calculations than the direct computation of the SKT condition. On the other hand, note that Theorem 5.1 has a wide range of applications, since it requires no hypothesis on the base complex manifold.