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, who introduced it in 1964.
Definition.
For a complex manifold formula_1, its "Aeppli cohomology" is given by:
 formula_2
formula_3 and formula_4 denote the Dobeault operators.
Maps.
de Rham and Dobeault cohomology are given by:
 formula_5
 formula_6
 formula_7
Since there are canonical inclusions formula_8 and formula_9, there is a canonical inclusion of de Rham into Aeppli cohomology:
 formula_10
Since there are canonical inclusions formula_11 as well as formula_12 and formula_13, there are canonical maps from Dobeault into Aeppli cohomology:
 formula_14
 formula_15
Furthermore there are canonical maps formula_16 from Bott–Chern cohomology, with all three compositions formula_17 being identical.