Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence

Abstract Serre’s duality theorem implies a symmetry between the Hodge numbers, hp,q = hn−p,n−q, on a compact complex n–manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies dimE1p,q=dimE1n−p,n−q \dim E_1^{p,q} = \dim E_1^{n - p,n - q} for all p, q. Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” dimEkp,q=dimEkn−p,n−q \dim E_k^{p,q} = \dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].


Introduction
Associated to a complex manifold one has the so-called Frölicher or Hodge-to-de Rham spectral sequence [7] which relates the Dolbeault cohomology to the de Rham cohomology of the manifold. As a consequence, one obtains relations between the dimensions of the Dolbeault cohomology and topological invariants of the manifold in the compact case. Frölicher [7] observes the inequality p+q=k h p,q ≥ b k , where h p,q denotes the dimension of the Dolbeault cohomology group H p,q ∂ , and b k is the k th Betti number, along with the equality p,q (− ) p+q h p,q = i (− ) i b i , with the right-hand term being the topological Euler characteristic. The inequality becomes an equality for all k if and only if the spectral sequence degenerates on its rst page, which happens e.g. if the complex structure is Kähler or more generally, Moišezon (see e.g. [6]). In the past few decades there has been interest in studying examples of compact complex manifolds for which the Frölicher spectral sequence degenerates only on the second page or later (see e.g. [2], [5]), with complex nilmanifolds being most amenable to this study. In this short note, we will observe through an immediate adaptation of the arguments in [3] that the symmetry dim E p,q = dim E n−p,n−q we have on the rst page of the spectral sequence, due to Serre duality on a compact complex n-manifold, continues to hold on all subsequent pages, thereby signi cantly reducing the amount of computation required to obtain these dimensions. This result has been known to Stelzig from his study of double complexes [10] and Popovici via harmonic theory (see Popovici-Stelzig-Ugarte [9, Remark 2.10]); the current author wishes to illustrate that with Serre duality in hand, the result follows by a very quick algebraic argument. We note that this result shortens some of the Frölicher spectral sequence calculations for a hypothetical complex S done in [1]. The argument presented here for the persistence of Serre symmetry also extends to situations in which one does not have an apparent double complex, for example for the Dolbeault cohomology of almost complex manifolds studied by Cirici and Wilson [4] (see Corollary 2.8 and Remark 2.9).

Serre symmetry
De nition 2.1. (cf. [3]) An n-dimensional bigraded Poincaré ring is a bigraded ring A with the following properties: (1) Each A p,q is a nite-dimensional complex vector space, and there exists an n such that A p,q = { } if p > n or q > n and A n,n is one-dimensional. (2) Denote A k = p+q=k A p,q . If α ∈ A k and β ∈ A l then αβ = (− ) kl βα.
(3) Fix a nonzero element ξ ∈ A n,n . De ne a bilinear pairing A p,q ⊗ A n−p,n−q −,− −→ C by α, β ξ = αβ. We require the linear map in−p,n−q from A n−p,n−q to A p,q* (the complex-linear dual of A p,q ) which sends β ∈ A n−p,n−q to the functional −, β to be an isomorphism for all p, q. (Note that this property does not depend on the choice of nonzero ξ .) Example 2.2. For X a compact complex manifold of complex dimension n, the Dolbeault cohomology H *,* ∂ is a bigraded Poincaré ring. Indeed, the space of (p, q) forms for p, q not between and n is trivial; each H p,q ∂ is nite-dimensional by elliptic theory; H n,n ∂ is one-dimensional, spanned by a volume form for the manifold. Therefore property (1) of the de nition is satis ed. Property (2) follows from the same property holding on the algebra of all complex-valued di erential forms on X and the fact that∂ is a derivation with respect to the product of forms. As for property (3), let ξ = [Ω]∂ where Ω is some xed (n, n) volume form on X. Sincē ∂Ω = and it is not∂-exact (since otherwise it would be d-exact for degree reasons, and this cannot be by Stokes' theorem), ξ spans H n,n ∂ . Now, we will show that in−p,n−q is an isomorphism; since by Serre duality dim H p,q ∂ = dim H n−p,n−q ∂ , it will su ce to show injectivity. Take a non-zero α ∈ H n−p,n−q ∂ and take the∂harmonic representative y of this class. Denote by *y the conjugate of its Hodge dual, i.e. the (p, q)-form such that y ∧ *y = y, y L Ω. Since *∂*y = −∂ * y = , we have ∂*y = and hence∂*y = . Now y ∧ *y = ||y|| L Ω, and ||y|| L ≠ , and thus passing to cohomology we obtain that

De nition 2.3. (cf. [3]) A di erential on an n-dimensional bigraded Poincaré ring A is a linear map
Example 2.4. For X a compact complex manifold, the induced di erential ∂ on A = H∂(X) (given by ∂[x]∂ = [∂x]∂) is a di erential of bidegree ( , ) on the bigraded Poincaré ring H∂(X) in the above sense. Indeed, property (i) is satis ed, and properties (ii) and (iii) are satis ed by ∂ on the level of forms (note that the derivation ∂ on forms induces a derivation on H∂(X)). As for property (iv), we note that A n− = A n− ,n ⊕A n,n− . The di erential vanishes on A n,n− for degree reasons. Now take [x]∂ ∈ A n− ,n and suppose ∂[x]∂ = [∂x]∂ is non-zero. Since A n,n is one-dimensional, this means there is a non-zero constant c such that ∂x − cΩ =∂y for some (n, n − )-form y. Rearranging, we get Ω = ∂( x c ) −∂( y c ). For degree reasons, this is the same as Ω = d( x−y c ), which by Stokes' theorem would imply X = ∅. (Alternatively, the vanishing of this di erential follows from the convergence of the Frölicher spectral sequence to the complexi ed de Rham cohomology and H n dR (X; C) ∼ = C.)

Proposition 2.5. (cf. [3]) The cohomology ring HA = ker d/ im d of an n-dimensional bigraded Poincaré ring A with di erential d is an n-dimensional bigraded Poincaré ring.
Proof. Since d has a well-de ned bidegree we obtain a decomposition HA = p,q HA p,q , where HA p,q = (ker d ∩ A p,q )/ im d. Note that property (iv) implies that HA inherits properties (1) and (2). If ξ is the (n, n)-bidegree element used to de ne −, − on A, then we can de ne a bilinear pairing HA p,q ⊗ HA n−p,n−q −,− −→ C on the cohomology ring by [ (3), i.e. that i n−p,n−q is an isomorphism for all p, q. Denote the bidegree of d by (p , q ); recall p + q = . For α ∈ A p,q and β ∈ A n−p−p ,n−q−q , by properties (iii) and (iv) of the di erential, we have = d(αβ) = (dα)β + (− ) p+q α(dβ). From the de nition of −, − , this implies dα, β = (− ) p+q− α, dβ . Consider the following diagram (extending to left and right): Here d * denotes the dual morphism to d; we have d * ( −, β ) = d−, β . Note that the calculation preceding the diagram implies that i is a chain map for all p, q. Indeed, let us check commutativity for the left square: for β ∈ A n−p−p ,n−q−q we have in−p,n−q(dβ) = −, dβ , while Since in−p,n−q is an isomorphism for all p, q by property (3), it follows that the chain map i is a quasiisomorphism. We would now like to relate this induced map to the map i n−p,n−q we wish to establish bijectivity for.
First of all, for β ∈ ker d ∩ A n−p,n−q , we have in−p,n−q(β) = −, β , and so the induced map i * n−p,n−q sends [β] to the d * -cohomology class [ −, β ]. Now we note that the target space ker d * ∩ (A p,q ) * / im d * is not quite (HA p,q ) * = (ker d ∩ A p,q / im d) * . Regardless, by the universal coe cient theorem, the map Ψ from the former to the latter given by Ψ(  Recall that for a complex manifold X, denoting its algebra of complex-valued smooth forms by A •,• (X), the Frölicher spectral sequence is associated to the ltration F p A •,• (X) = i≥p,j A i,j (X). The complexi ed de Rham di erential d preserves this ltration, d(F p ) ⊂ F p , and if x ∈ F p , y ∈ F q , then x ∧ y ∈ F p+q . Now by [8,Theorem 2.14], it follows that the di erential on the E k page of the Frölicher spectral sequence, for any k ≥ , satis es property (iii). We know that it also satis es properties (i) with bidegree (k, − k) and (ii). Property (iv) is trivially satis ed for degree reasons (or by the alternative argument as given in Example 4). In particular, (E , d ) is a bigraded Poincaré ring with di erential, and so inductively we have the following: Remark 2.9. It is as of yet unknown whether Serre symmetry holds in general for the Dolbeault cohomology of a (not necessarily integrable) almost complex structure on a closed manifold as studied in [4]. If these groups were known to be nite-dimensional satisfying Serre symmetry, then the symmetry would hold on all subsequent pages of the associated spectral sequence.