In complex geometry in mathematics, Aeppli cohomology is a cohomology theory for complex manifolds. It serves as a bridge between de Rham cohomology, which is defined for real manifolds which in particular underlie complex manifolds, and Dobeault cohomology, which is its analogue for complex manifolds. A direct comparison between these cohomology theories through canonical maps is not possible, but both canonically map into Aeppli cohomology. A similar cohomology theory, which maps into both and which hence also serves as a bridge is Bott–Chern cohomology. Aeppli cohomology is named after Alfred Aeppli (1928–2008), who introduced it in 1964.
Definition For a complex manifold , its Aeppli cohomology is given by:Aeppli 1964, p. 63Angella & Tomassini 2014, p. 1 & 1.1. Bott-Chern cohomologyAngella 2015, p. 5
and denote the Dobeault operators.
Maps de Rham and Dobeault cohomology are given by:Angella 2015, p. 3-4
Since there are canonical inclusions and , there is a canonical inclusion of de Rham into Aeppli cohomology:
Since there are canonical inclusions as well as and , there are canonical maps from Dobeault into Aeppli cohomology:
Furthermore there are canonical maps from Bott–Chern cohomology, with all three compositions being identical.
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