Global stability of the Pluriclosed flow on compact simply-connected simple Lie groups of rank two

We compute the (1,1)-Aeppli cohomology of compact simply-connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds. Moreover, we show that for compact simply-connected simple Lie groups of rank two the Dolbeaut cohomology, as well as the Bott-Chern and the Aeppli cohomologies, arise from just the left-invariant forms and we computed the whole Bott-Chern diamonds of SU(3) and Spin(5) when they are equipped with a left-invariant isotropic complex structure.


Introduction
Given a Hermitian manifold (X, J, g), the Bismut connection ∇ B associated to (g, J) is a Hermitian connection on X with totally skew-symmetric torsion, where by Hermitian connection we mean a connection which is compatible with both the metric and the complex structure, i.e. ∇g = ∇J = 0. It is described with respect to the Levi-Civita connection ∇ LC as, g ∇ B x y, z = g ∇ LC x y, z + 1 2 Jdω(x, y, z) , where ω is the Kähler form associated to g and J acts as Jdω(·, ·, ·) = −dω(J·, J·, J·). Since the Levi-Civita connection is torsion free, the above formula is prescribing the torsion of this connection as T B (x, y, z) = Jdω(x, y, z) . We indicate by Ric B the Ricci curvature tensor of the Bismut connection, which is Ric B (X, Y ) := tr g Ω B (X, Y ), i.e. the contraction of the endomorphism part of the Bismut curvature tensor Ω B (X, Y ) = [∇ X , ∇ Y ] − ∇ [X,Y ] . We also use Ric B to indicate the Ricci form of the Bismut connection which satisfies the following equation (see for example [6] and the references therein), Ric B (g) = − √ −1∂∂ log ω n − dd * g ω . In the equality above, d * g = ∂ * g + ∂ * g where ∂ * g : ∧ p+1,q X → ∧ p,q X and ∂ * g : ∧ p,q+1 X → ∧ p,q X are the L 2 g -adjoint operators of ∂ and ∂ respectively. In particular, the (1, 1)-component of this form is In [12] Streets and Tian introduced the pluriclosed flow with the intent to study the complex geometry of Hermitian non-Kähler manifolds. This is a parabolic flow of Hermitian metrics defined as follows. On a Hermitian manifold (X, g 0 , J) it evolves the metric obeying the equation: ∂ ∂t g = −S + Q g(0) = g 0 where S is the trace in the first two entries of the Chern curvature tensor Ω Ch and Q is a given quadratic polynomial in the torsion T Ch of the Chern connection. More precisely, S ij = (T r g Ω Ch ) ij = g kl Ω Ch klij , and Q ij = g kl g mn T Ch ikn T Ch jlm . In the literature, a Hermitian metric is said to be pluriclosed or SKT when its associated Kähler form ω satisfies dd c ω = 0. It can be proved that the pluriclosed flow preserves this condition whenever it occurs. Indeed, it evolves a pluriclosed metric in the direction of the (1, 1)-component of its Bismut Ricci form, which is dd c -closed (see (1.1)). More precisely, given a complex manifold (X, J) together with a pluriclosed metric ω 0 the pluriclosed flow evolves as Moreover, in [4,Theorem 9] it is proved that metrics with vanishing Bismut curvature, called Bismut flat metrics, are pluriclosed. Thus, clearly, Bismut flat metrics are fixed points for the pluriclosed flow.
Recently, in [8] the authors implemented a beautiful machinery based on Generalized Geometry to compare metrics with the same torsion class, which, for a generic metric ω, is the class of [∂ω] ∈ H 2,1 ∂ . Thanks to it, they proved that the Bismut flat metrics are attractive for the pluriclosed flow in their torsion class. Precisely, they proved the following theorem. Theorem 1.1 (Theorem 1.2 of [8]). Let (X 2n , J, ω BF ) be a compact Bismut flat manifold. Given ω 0 a pluriclosed metric such that [∂ω 0 ] = [∂ω BF ] ∈ H 2,1 ∂ (X), the solution to pluriclosed flow with initial data ω 0 exists on [0, ∞) and converges to a Bismut flat metric ω ∞ .
It is conjectured that the SKT metrics with vanishing Bismut Ricci form should have a similar behaviour, however, this is the first result showing that a natural class of non-Kähler metrics is attractive for this flow. We fix here the notation for the Bott-Chern and Aeppli cohomologies. The Bott-Chern cohomology, [7], is the Z 2 -graded algebra while the Aeppli cohomology, [1], is the both with respect to the double complex ∧ •,• X, ∂, ∂ . Theorem 1.1 together with the knowledge of the (1, 1)-Aeppli cohomology of the manifold lead to global stability results. For example, the Hopf surface U(1) × SU(2) and the Calabi-Eckman manifold SU(2) × SU(2) are known to be Bismut flat when they are equipped with their standard complex structures and the metrics ω BF 's coming from the Killing forms; moreover, in both cases the (1, 1)-Aeppli cohomology is one-dimensional and generated by the class of ω BF , see Theorem 3.3 and Proposition 3.4 of [5]. Given any pluriclosed metric ω on them, integrating over the fiber we have that Theorem 1.1 applies giving convergence of the pluriclosed flow to a Bismut-flat structure for any initial pluriclosed data.
In this note we produce new examples of global stability for the pluriclosed flow other than on Calabi-Eckmann manifolds. We consider compact simply-connected simple Lie groups of rank two equipped with their isotropic complex structures (as defined in Section 3) and the Hermitian metrics coming from the Killing forms ω BF 's. These are classified as SU (3), Spin(5) and G 2 (see Remark 2.3). Moreover, they are Bismut flat and we prove that their (1, 1)-Aeppli cohomologies are . We performed these computations combining the results of [3] with the presentation of the Dolbeaut cohomology for compact simply-connected simple Lie group of rank two given in [10]: where subscripts denote bi-degree of the generators x, y, u and the isotropic complex structures are the ones that together with the Killing metric give a Hermitian structure on X. We thus prove the following result.
Theorem For completeness, we compute the whole Hodge diamond for the Bott-Chern cohomology of SU(3) and Spin(5) when they are equipped with their isotropic complex structures (in Section 4). In particular, we prove that for compact simply-connected simple Lie groups of rank two the whole Dolbeaut, Bott-Chern and Aeppli cohomologies arise from just the left-invariant classes and we have the following Bott-Chern numbers, respectively for SU(3) and Spin(5): A direct computation can also be performed to compute the Bott-Chern numbers of G 2 .

Complex structures on Bismut flat manifolds
In [11] Samelson showed (by an explicit construction) that any even-dimensional compact Lie group admits a left-invariant complex structure compatible with the bi-invariant metric coming from the Killing form.
In [2] Alexandrov and Ivanov showed that any even dimensional connected Lie group equipped with a bi-invariant metric g and a left-invariant complex structure which is compatible with g is Bismut flat. Afterwords, in [13] the authors showed that up to taking the universal cover, these are the only existing Bismut flat manifolds. In other words, simply-connected Bismut flat manifolds have been characterized as Samelson spaces, which definition is as follows.  , J), where G is a connected and simply-connected, even-dimensional Lie group, g a bi-invariant metric on G, and J a leftinvariant complex structure on G that is compatible with g. By Milnor's Lemma (Lemma 7.5 of [9]), a simply-connected Lie group G with a bi-invariant metric must be the product of a compact semisimple Lie group with an additive vector group.
Lemma 2.2 (Lemma 7.5 of [9]). Let G be a simply-connected Lie group with a bi-invariant metric ·, · . Then G is isomorphic and isometric to the product G 1 × · · · × G r × R k where each G i is a simply-connected compact simple Lie group and R k is the additive vector group with the flat metric.
We can go even further in the classification of the Bismut flat manifolds; indeed, the compact simply-connected simple Lie groups are fully classified, they are: (5) and G 2 are the only compact simply-connected simple Lie groups of rank two. Hence, the Dolbeaut cohomology of these manifolds can be computed using Pittie's cohomology presentation (1.2).
In [10] Pittie gave a complete description of the moduli of left-invariant, integrable complex structures on even-dimensional compact Lie groups, proving that they all come from Samelson's construction in [11], namely, from a choice of a maximal torus, a complex structure on the Lie algebra of the torus, and a choice of positive roots for the Cartan decomposition. Let us now recall this construction with more details.
In the following let G be an even dimensional connected Lie group equipped with a bi-invariant metric ·, · and denote by g the Lie algebra of G and by g C its complexification. ·, · will also denote the induced inner product on g.
We know that the left-invariant complex structures on G are linear maps J : g → g such that JY ] for any X, Y in g. These are in one to one correspondence with the complex Lie subalgebras s ⊂ g C , such that s, s = 0, s ∩ g = 0, and s ⊕ s = g C . Such subspaces are called Samelson subalgebras of g C .
Given K a maximal torus of G, it is well-known that one has the ad(K)-invariant decomposition where k C denotes the complexification of k, R + is the space of positive roots and Since dim(G) is even, we know that the abelian Lie algebra k is even dimensional. So we can choose an almost complex structure on k that is compatible with the metric. This is equivalent to choose a complex subalgebra a ⊂ k C such that a, a = 0, a ∩ k = 0, and a ⊕ a = k C . Now one could simply take to be the Samelson subalgebra. Pittie [10] proved that, any left-invariant complex structure on G is obtained this way. Moreover, he described the moduli space of left-invariant complex structures on G as m 2 (G) = (GL(2k, R)/GL(k, C)) /F , where k is the rank of G and F is a discrete group generated by the automorphisms of the abelian factor of G, the automorphisms of the Dynkin diagrams of the simple factors, and permutations among isomorphic simple factors. Here m 2 (G) is the space of left-invariant complex structures on G up to automorphisms of G. Among these, the isotropic ones are given by the quotient by F of O(2k)/U (k).
As a consequence, the moduli space of left-invariant complex structures of a compact simplyconnected rank two simple Lie group X is given by 3. The isotropic complex structure on compact simply-connected simple Lie groups of rank two In this section we follow the Samelson construction to describe the left-invariant complex structures on compact simply-connected simple Lie groups of rank two. We are going to focus on the complex structures which are isotropic with respect to the Killing metric. In the following ·, · will denote the inner product on the algebra induced by the Killing form. Since the only compact simply-connected simple Lie groups of rank two are SU(3), Spin(5) and G 2 (see Remark 2.3) we split this section in three parts. However, we may use the same symbols for objects which refer to different groups.
By differentiating this two conditions we get that its Lie algebra is made by the skew-Hermitian for real parameters a, b, c, d, e, f, g, h. We can take i-times the Gell-Mann matrices as basis of the SU(3) Lie algebra su (3): Here λ ijk = (−1) |σ| λ σ(i,j,k) for any permutation σ. With this notation Thus we notice that all the e i have the same norm. The maximal torus in SU (3) is given by the diagonal matrices and its Lie algebra k ⊂ su(3) is generated by e 3 and e 8 . The remaining six generators, outside the Cartan subalgebra, could be arranged into six roots. Consider e 1 ± i e 2 , e 4 ± i e 5 and e 6 ± i e 7 ; then we have the following relations: 3.2. Spin (5). The group Spin (5) is the double cover of SO(5), hence they share the same Lie algebra, which is given by the 5 × 5 skew-symmetric matrices. We take the following generators as basis of the Spin(5) Lie algebra spin (5): where A i,j represents the 5 × 5 skew-symmetric matrix with 1 in the (i, j)-position; more precisely, (A i,j ) p,q = δ i,p δ j,q − δ i,q δ j,p . Using this notation we describe the structure constants of the global left-invariant frame e 1 , e 2 , e 3 , e 4 , e 4 , e 5 , e 6 , e 7 , e 8 , e 9 , e 10 on Spin (5) as A maximal torus in SO (5)  Therefore, we have the eight roots ±(i, 0), ±(0, i), ±(i, i), ±(i, − i). A choice of positive roots leads to a Samelson subalgebra of Spin(5) and we can take generators ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 , ϕ 5 defined as The two isotropic Samelson subalgebras are detected by the choices a + i b = ± i and we indicate the corresponding complex structures with J ± . Indeed, it can be verified that e 1 , e 10 = 0 and they have the same norm.

Bott Chern cohomology of compact simply-connected simple Lie groups of rank two
We now compute case by case the (1, 1)-Aeppli cohomology of the compact simply-connected simple Lie groups of rank two in order to apply Theorem 1.1.
We will use the following notation: ϕ ij = ϕ i ∧ ϕ j .

SU(3).
We consider the Hermitian pluriclosed manifold (SU(3), J 0,−1 , ω BF ) where J 0,−1 is the isotropic left-invariant complex structure as given in Section 3.1 and ω BF represents the Hermitian metric coming from the Killing form; more precisely it is By computing the complex structure equations, we obtain In [10], a model for the Dolbeault cohomology of compact simply-connected simple Lie groups of rank two is given, see (1.2). In particular, when SU (3) is equipped with its isotropic left-invariant complex structure it holds where subscripts denote bi-degree. By this presentation we recover that the Hodge numbers are Consider the sub-complex of left-invariant forms ι : Since it is closed for the C-linear Hodge- * -operator * g BF : ∧ • 1 ,• 2 X → ∧ n−• 2 ,n−• 1 X associated to g BF , H ∂ (ι) is injective, see Theorem 1.6 and Remark 1.9 of [3]. By knowing the Hodge numbers, we want to prove that it is also surjective. Thanks to structure of the model (4.2) it is enough to check that the sub-complex cohomology has the same dimension as the complex in bi-degree (0, 1), (1, 1) and (2,1). That is, to verify that the sub-complex has cohomologies H 0,1 (3)) inv of dimension one. We shall highlight that we chose the subcomplex of the left-invariant forms to exploit its algebraic structure. Indeed, the Pittie model induces the following sub-complex ι : x 0,1 , x 0,1 , y 1,1 , y 1,1 , u 2,1 , which, by construction, verifies H ∂ (ι ) surjective. However, it is not trivial to prove that it closed for the C-linear Hodge- * -operator associated to any metric, and thus it is difficult to prove that H ∂ (ι ) is also injective. A direct computation leads to the following conditions Therefore, H ∂ (ι) is an isomorphism and the formal representative x 0,1 , y 1,1 and u 2,1 of Pittie's model are respectively in the left-invariant classes In particular, we get that ⊕C ϕ 1234123 ⊕ C ϕ 12341234 .

By Theorem 1.3 and Proposition 2.2 in [3], we also have that H BC (ι) is an isomorphism. In particular, the Bott-Chern cohomology arises from just the left-invariant forms and a direct computation leads to
Hence, the Bott-Chern numbers are We know that for a Hermitian metric g on a complex manifold X, the C-linear Hodge- * -operator induces the isomorphism * g BF : Hence, as a corollary, we get that dim H 1,1 A (SU(3)) = 1. In particular, we have that H 1,1 A (SU(3)) = C ϕ 11 + ϕ 22 + ϕ 33 + ϕ 44 = C [ω BF ] , thus thanks to Theorem 1.1 the following result holds.
where ω BF is the metric given by the Killing form (as in (4.1)) and J 0,−1 is the isotropic left-invariant complex structure defined in Section 3.1. Given any pluriclosed metric ω 0 on (SU(3), J 0,−1 ) the solution to the pluriclosed flow with initial data ω 0 exists on [0, ∞) and converges to a Bismut flat metric ω ∞ . In particular, there exists a positive λ such that (3)) and ω ∞ = λ ω BF .
Proof. First of all, we notice that the complex structure J 0,−1 is such that the maximal torus We have seen that Bismut flat metrics are bi-invariant and it is well known that any invariant symmetric bi-linear form on a simple Lie group must be a multiple of the Killing form. Thanks to the Milnor result (Lemma 2.2) we know that there is only one Lie group structure on SU(3) (as manifold) which may admit a bi-invariant metric. Thus ω ∞ must be a positive multiple of ω BF , and hence ω ∞ = λ ω BF . (5). We now consider the Hermitian pluriclosed manifolds (Spin(5), J ± , ω BF ) where J ± are the only two isotropic left-invariant complex structure on Spin(5) as described in Section 3.2 and ω BF represents the Hermitian metric coming from the Killing form, which is
Therefore, H ∂ (ι) is an isomorphism and we get that the Dolbeaut cohomology ring is As before, applying Theorem 1.3 and Proposition 2.2 of [3] we obtain that H BC (ι) is an isomorphism. In particular, the Bott-Chern cohomology arises from just the left-invariant forms as well as the Aeppli cohomology. A direct computation shows that dim H 1,1 A (Spin(5)) C and it is generated by the class of ω BF , that is   (5), ω BF , J ± ) where ω BF is the metric given by the Killing form (as in (4.3)) and J ± are the isotropic left-invariant complex structure defined in Section 3.2. Given any pluriclosed metric ω 0 on (Spin(5), J ± ) the solution to the pluriclosed flow with initial data ω 0 exists on [0, ∞) and converges to a Bismut flat metric ω ∞ . In particular, there exists a positive λ such that [ω 0 ] = λ[ω BF ] in H 1,1 A (Spin (5)) and ω ∞ = λ ω BF .
For completeness, we compute the whole Bott-Chern cohomology ring, which is Hence, the Bott-Chern numbers are Finally, we consider the Hermitian pluriclosed manifolds (G 2 , J ± , ω BF ) where J ± are the only two isotropic left-invariant complex structure on G 2 as described in the Section 3.3 and ω BF represents the Hermitian metric coming from the Killing form, which is (4.4) ω BF := i 3ϕ 11 + ϕ 22 − 3ϕ 33 + 12ϕ 44 − 36ϕ 55 + 36ϕ 66 − 12ϕ 77 .
Theorem 4.3. Consider (G 2 , ω BF , J ± ) where ω BF is the metric given by the Killing form (as in (4.4)) and J ± are the isotropic left-invariant complex structure defined in Section 3.3. Given any pluriclosed metric ω 0 on (G 2 , J ± ) the solution to the pluriclosed flow with initial data ω 0 exists on