Frolicher Spectral SequenceEdit
The Frölicher spectral sequence is a central construction in complex geometry that links the Dolbeault cohomology of a complex manifold to its de Rham cohomology. It emerges from the natural double complex of smooth differential forms on a complex manifold X, equipped with the two exterior differential operators ∂ and ∂̄. On a compact complex manifold, this double complex provides a systematic way to organize information about holomorphic and anti-holomorphic directions and to compare it with the global topology captured by de Rham cohomology.
Concretely, one starts with the bigraded space of smooth forms A^{p,q}(X) and the two differentials ∂: A^{p,q} → A^{p+1,q} and ∂̄: A^{p,q} → A^{p,q+1}. The total differential d = ∂ + ∂̄ yields the de Rham complex (A^{•,•}(X), d), whose cohomology is the de Rham cohomology H^k_{dR}(X). Filtering the total complex by the holomorphic degree p gives rise to the Frölicher spectral sequence with pages E_r^{p,q}, starting at E1^{p,q} ≅ H^{q}(X, Ω^p) (the Dolbeault cohomology groups). The differentials d_r go from E_r^{p,q} to E_r^{p+r,q−r+1}, encoding how the ∂-direction interacts with the ∂̄-direction beyond first order. As r grows, the sequence stabilizes and yields an associated graded object for the total cohomology: Gr^F H^{n}{dR}(X) ≅ ⊕{p+q=n} E∞^{p,q}.
The framing is cleanly expressed through the language of spectral sequences and double complexes. In particular, the first page E1 is built from the Dolbeault cohomology groups, E1^{p,q} ≅ H^{q}(X, Ω^p), and subsequent pages refine this data by measuring obstructions to assembling the local holomorphic information into global forms. The whole construction is functorial in families and serves as a diagnostic for the relationship between the complex structure of X and its topology.
Construction and basic properties
- The starting point is the Double complex (A^{p,q}(X), ∂, ∂̄) of smooth (p,q)-forms on X. The total complex (⊕_{p+q=k} A^{p,q}(X), d) computes de Rham cohomology.
- The Frölicher spectral sequence arises from the natural filtration by p, yielding pages E_r^{p,q} with differentials d_r: E_r^{p,q} → E_r^{p+r,q−r+1}.
- On a compact complex manifold, the sequence converges to H^{p+q}{dR}(X), with the relation Gr^F H^{n}{dR}(X) ≅ ⊕_{p+q=n} E∞^{p,q}. This connects the global topology to the holomorphic and anti-holomorphic structures.
- The first page is Dolbeault-cohomology-based: E1^{p,q} ≅ H^{q}(X, Ω^p). The higher pages detect how the holomorphic p-forms fail to assemble into global closed forms after accounting for ∂-growth.
These ideas are often illustrated through the standard references to Dolbeault cohomology and de Rham cohomology within the context of complex geometry complex manifolds.
Degeneration and the Kähler case
A particularly important situation is when X is a compact Kähler manifold. In that case, the Frölicher spectral sequence degenerates at E1, meaning all higher differentials d_r for r ≥ 1 vanish. Equivalently, the de Rham cohomology admits a Hodge-type decomposition H^{n}{dR}(X) ≅ ⊕{p+q=n} H^{q}(X, Ω^p), and the dimensions h^{p,q} := dim H^{q}(X, Ω^p) (the Hodge numbers) control the entire cohomology. This degeneration is a manifestation of Hodge theory, which ties the topology of X directly to its complex structure via inner products on differential forms and harmonic representatives Hodge theory.
In the non-Kähler setting, the situation is subtler. The spectral sequence may not degenerate at E1, and nontrivial differentials can persist on higher pages. These differentials measure obstructions to a Hodge-type decomposition and reflect subtle interactions between the complex structure and the differential geometry of X. The failure of degeneration can be used to distinguish complex manifolds that are not Kähler from those that are, highlighting the separation between topological invariants and complex-analytic structure.
Non-Kähler phenomena and examples
- Hopf surfaces and other non-Kähler complex surfaces provide concrete examples where the Frölicher spectral sequence does not degenerate at E1. In such cases, the Dolbeault data alone is not sufficient to reconstruct the full de Rham cohomology, and higher-order differentials contribute nontrivially.
- The Iwasawa manifold, a nilmanifold with a left-invariant complex structure, is another classic example where the Frölicher spectral sequence exhibits nontrivial higher-page behavior, illustrating how non-Kähler geometry can reveal richer spectral structures than the Kähler case.
- Frölicher’s inequality expresses a general relationship between the Betti numbers b_k = dim H^{k}_{dR}(X) and the Hodge numbers h^{p,q} = dim H^{q}(X, Ω^p). It provides a numerical constraint on how far a given complex manifold can be from satisfying a full Hodge decomposition, with equality in the Kähler case.
These examples emphasize that the Frölicher spectral sequence is not merely a formal gadget but a diagnostic tool for the interplay between topology and complex structure on a manifold. See also Hopf surface and Iwasawa manifold for detailed expositions of these non-Kähler examples, and Kähler manifold for the contrast with the degenerating case.
Historical context and development
The spectral sequence bearing Frölicher’s name was introduced to organize the passage from local holomorphic data to global topological information on complex manifolds. It provides a bridge between the Dolbeault cohomology that captures holomorphic content and the global de Rham cohomology that encodes topology. The theory has since become a standard tool in complex geometry, influencing investigations into deformation theory, non-Kähler geometry, and the study of special metrics on complex manifolds.
Key aspects of the theory are developed in the language of filtrations on cohomology, the behavior of differentials on higher pages, and the interpretation of E∞ as the associated graded object to a natural filtration on de Rham cohomology. The framework interacts with broader themes in global analysis and algebraic topology, including the analysis of elliptic operators and the role of harmonic representatives when a rich metric structure is available.