nLab Frölicher spectral sequence

Idea

The Frölicher spectral sequence is the spectral sequence of a double complex of the Dolbeault double complex $(\Omega^{\bullet,\bullet}, \partial, \bar \partial)$ of a complex analytic space $X$. It converges to the complex de Rham cohomology and hence may be thought of as interpolating between that and Dolbeault cohomology. In particular on a compact Kähler manifold $X$ the Frölicher spectral sequence degenerates on the $E_1$-page and exhibits the Hodge filtration (see here) on its complex de Rham cohomology. Therefore the higher differentials of the Frölicher spectral sequence may be thought of as witnessing obstructions to $X$ admitting a Kähler structure.

Write $\Omega^\bullet$ for the holomorphic de Rham complex. The Frölicher spectral sequence has the form

$$E_1^{p,q} = (H^q(X,\Omega^p), \partial) \Rightarrow H^{p+q}(X, \Omega^\bullet)$$

with $E_\infty^{p,q} = Gr^p H^{p+q}(X)$ the associated graded of the Hodge filtration

$$F^p H^k(X,\mathbb{C}) \coloneqq im\left( H^k(X, \Omega^{\bullet \geq p}) \to H^k(X, \Omega^\bullet) \right) \,.$$

Theorem

If $X$ is compact and carries the structure of a Kähler manifold, then the Frölicher spectral sequence degenerates at $E_1$.

Traditionally this was proven (Frölicher 55) by way of harmonic differential forms. In (Deligne-Illusie 87) this is proven instead by reduction to positive characteristic.

Related concepts

• Hodge-de Rham spectral sequence

References

Original articles include

• Alfred Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proceedings of the National Academy of Sciences 41: 641–644 (1955)

• Pierre Deligne, Luc Illusie Relèvements modulo $p^2$ et decomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247-270.

Further developments include

• Laura Bigalke, Sönke Rollenske, The Frölicher spectral sequence can be arbitrarily non degenerate, Math. Ann., 341(3), 623–628, 2008 (arXiv:0709.0481)

A textbook account is in

• Claire Voisin, section 8.3.3 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Claire Voisin, Hodge theory and the topology of compact Kähler and complex projective manifolds (pdf)