Aeppli cohomology

Template:Short description 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 $X$, its Aeppli cohomology is given by:

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

$\partial$ and $\bar{\partial }$ denote the Dobeault operators.

Maps

de Rham and Dobeault cohomology are given by:

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

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

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

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

$H_{\mathrm{d}\mathrm{R}}^{p+q}(X)\to H_{\mathrm{A}}^{p,q}(X).$

Since there are canonical inclusions $\mathrm{img}({\partial }_{p-1,q}),\mathrm{img}({\bar{\partial }}_{p,q-1})\hookrightarrow \mathrm{img}({\partial }_{p-1,q})+\mathrm{img}({\partial }_{p,q-1})$ as well as $\ker ({\partial }_{p,q})\hookrightarrow \ker ({\bar{\partial }}_{p+1,q}{\partial }_{p,q})$ and $\ker ({\bar{\partial }}_{p,q})\hookrightarrow \ker ({\partial }_{p,q+1}{\bar{\partial }}_{p,q})$, there are canonical maps from Dobeault into Aeppli cohomology:

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

$H_{\bar{\partial }}^{p,q}(X)\to H_{\mathrm{A}}^{p,q}(X).$

Furthermore there are canonical maps $H_{\mathrm{B}\mathrm{C}}^{p,q}(X)\to H_{\mathrm{d}\mathrm{R}}^n(X),H_{\partial }^{p,q}(X),H_{\bar{\partial }}^{p,q}(X)$ from Bott–Chern cohomology, with all three compositions $H_{\mathrm{B}\mathrm{C}}^{p,q}(X)\to H_{\mathrm{A}}^{p,q}(X)$ being identical.

Literature

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