Aeppli cohomology

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.

For a complex manifold ${\displaystyle X}$, its Aeppli cohomology is given by:^[1][2][3]

${\displaystyle H_{\mathrm {A} }^{p,q}(X):=\ker(\partial _{p,q}{\overline {\partial }}_{p,q})/(\operatorname {img} (\partial _{p-1,q})+\operatorname {img} (\partial _{p,q-1})).}$

${\displaystyle \partial }$ and ${\displaystyle {\overline {\partial }}}$ denote the Dobeault operators.

de Rham and Dobeault cohomology are given by:^[4]

${\displaystyle H_{\mathrm {dR} }^{n}(X):=\ker(\mathrm {d} _{n})/\operatorname {img} (\mathrm {d} _{n-1}),}$

${\displaystyle H_{\partial }^{p,q}(X):=\ker(\partial _{p,q})/\operatorname {img} (\partial _{p-1,q}),}$

${\displaystyle H_{\overline {\partial }}^{p,q}(X):=\ker({\overline {\partial }}_{p,q})/\operatorname {img} ({\overline {\partial }}_{p,q-1}).}$

Since there are canonical inclusions ${\displaystyle \ker(\mathrm {d} _{p+q})\hookrightarrow \ker({\overline {\partial }}_{p+1,q}\partial _{p,q})=\ker(\partial _{p,q+1}{\overline {\partial }}_{p,q})}$ and ${\displaystyle \operatorname {img} (\partial _{p-1,q})+\operatorname {img} (\partial _{p,q-1})\hookrightarrow \operatorname {img} (\mathrm {d} _{p+q-1})}$, there is a canonical inclusion of de Rham into Aeppli cohomology:^[2]

${\displaystyle H_{\mathrm {dR} }^{p+q}(X)\rightarrow H_{\mathrm {A} }^{p,q}(X).}$

Since there are canonical inclusions ${\displaystyle \operatorname {img} (\partial _{p-1,q}),\operatorname {img} ({\overline {\partial }}_{p,q-1})\hookrightarrow \operatorname {img} (\partial _{p-1,q})+\operatorname {img} (\partial _{p,q-1})}$ as well as ${\displaystyle \ker(\partial _{p,q})\hookrightarrow \ker({\overline {\partial }}_{p+1,q}\partial _{p,q})}$ and ${\displaystyle \ker({\overline {\partial }}_{p,q})\hookrightarrow \ker(\partial _{p,q+1}{\overline {\partial }}_{p,q})}$, there are canonical maps from Dobeault into Aeppli cohomology:^[2]

${\displaystyle H_{\partial }^{p,q}(X)\rightarrow H_{\mathrm {A} }^{p,q}(X),}$

${\displaystyle H_{\overline {\partial }}^{p,q}(X)\rightarrow H_{\mathrm {A} }^{p,q}(X).}$

Furthermore there are canonical maps ${\displaystyle H_{\mathrm {BC} }^{p,q}(X)\rightarrow H_{\mathrm {dR} }^{n}(X),H_{\partial }^{p,q}(X),H_{\overline {\partial }}^{p,q}(X)}$ from Bott–Chern cohomology, with all three compositions ${\displaystyle H_{\mathrm {BC} }^{p,q}(X)\rightarrow H_{\mathrm {A} }^{p,q}(X)}$ being identical.

• Aeppli, Alfred (1964). "On the Cohomology Structure of Stein Manifolds". Proceedings of the Conference on Complex Analysis. Vol. 114. Springer, Berlin, Heidelberg. pp. 58–70. doi:10.1007/978-3-642-48016-4_7. ISBN 978-3-642-48016-4. {{cite book}}: ISBN / Date incompatibility (help)
• Angella, Daniele; Tomassini, Adriano (2014-11-21). "On Bott-Chern cohomology and formality". Journal of Geometry and Physics. 93: 52. arXiv:1411.6037. Bibcode:2015JGP....93...52A. doi:10.1016/j.geomphys.2015.03.004.
• Angella, Daniele (2015-07-25). "On the Bott-Chern and Aeppli cohomology". arXiv:1507.07112 [math.CV].