Frölicher spectral sequence - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold. Filter its complex-valued de Rham complex by holomorphic degree in the decomposition $\Omega^k(X;\mathbb C)=\bigoplus_{p+q=k}\Omega^{p,q}(X)$ . The resulting Frölicher spectral sequence has

E_1^{p,q}\cong H_{\bar\partial}^{p,q}(X),\qquad d_r:E_r^{p,q}\longrightarrow E_r^{p+r,q-r+1},

and converges to $H_{\mathrm{dR}}^{p+q}(X;\mathbb C)$ with its induced filtration. Its first page is therefore Dolbeault cohomology, while its limiting page gives the associated graded pieces of complex de Rham cohomology. Convergence alone does not supply a canonical direct-sum decomposition of de Rham cohomology.

Construction and early pages

The identities $d=\partial+\bar\partial$ , $\partial^2=\bar\partial^2=0$ , and $\partial\bar\partial+\bar\partial\partial=0$ make the spaces of $(p,q)$ -forms a double complex. Taking $\bar\partial$ -cohomology first gives $E_1$ , and $d_1$ is induced by $\partial$ . Higher differentials record successive obstructions to correcting a representative by forms of increasing holomorphic degree. Frölicher's original construction relates these pages to topological invariants Frölicher, pp. 641–644.

Degeneration and the Kähler case

The sequence degenerates at $E_r$ when all differentials from page $r$ onward vanish, so $E_r=E_\infty$ . If $X$ is a compact Kähler manifold, Hodge theory gives degeneration already at $E_1$ and identifies de Rham cohomology with the direct sum of its $(p,q)$ -pieces Voisin, §8.1. General compact complex manifolds need not have $E_1$ -degeneration.

Conventions and scope

Some sources call this the Hodge-to-de Rham or Hodge–de Rham spectral sequence. Indexing may begin at $E_0$ or $E_1$ , and decreasing versus increasing filtrations can reverse displayed bidegrees. The core uses the decreasing filtration by holomorphic degree and the cohomological differential convention shown there.

References

Alfred Frölicher, “Relations between the cohomology groups of Dolbeault and topological invariants,” Proceedings of the National Academy of Sciences 41 (1955), 641–644. DOI record. Relevant: construction and the resulting inequalities.
Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. DOI record. Relevant: §8.1, the Frölicher spectral sequence and Kähler degeneration.