nLab
Frölicher spectral sequence

Context

Complex geometry

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

The Frölicher spectral sequence is the spectral sequence of a double complex of the Dolbeault double complex (Ω ,,,¯)(\Omega^{\bullet,\bullet}, \partial, \bar \partial) of a complex analytic space XX. 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 XX the Frölicher spectral sequence degenerates on the E 1E_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 XX admitting a Kähler structure.

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

E 1 p,1=(H q(X,Ω p),)H p+q(X,Ω ) E_1^{p,1} = (H^q(X,\Omega^p), \partial) \Rightarrow H^{p+q}(X, \Omega^\bullet)

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

F pH k(X,)im(H k(X,Ω p)H k(X,Ω )). 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 XX is compact and carries the structure of a Kähler manifold, then the Frölicher spectral sequence degenerates at E 1E_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.

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 2p^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

Last revised on June 16, 2014 at 03:32:02. See the history of this page for a list of all contributions to it.