Bott–Chern cohomology

Short description: Cohomology theory for complex manifolds

In complex geometry in mathematics, Bott–Chern 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 Bott–Chern cohomology canonically maps into both. A similar cohomology theory, into which both map and which hence also serves as a bridge is Aeppli cohomology. Bott–Chern cohomology is named after Raoul Bott and Shiing-Chen Chern, who introduced it in 1965.

Definition

For a complex manifold $X$, its Bott–Chern cohomology is given by:

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

$\mathrm{d}$ denotes the exterior derivative while $\partial$ 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 is a canonical inclusion $\mathrm{img}({\partial }_{p-1,q}{\bar{\partial }}_{p-1,q-1})=\mathrm{img}({\bar{\partial }}_{p,q-1}{\partial }_{p-1,q-1})\hookrightarrow \mathrm{img}({\mathrm{d}}_{p+q})$, there is a canonical map from Bott–Chern cohomology into de Rham cohomology:

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

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

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

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

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

Literature

• Bott, Raoul; Chern, Shiing-Shen (1965). "Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections" (in en). Acta Mathematica 114: 71–112. doi:10.1007/BF02391818. 
• Angella, Daniele; Tomassini, Adriano (2014-11-21). "On Bott-Chern cohomology and formality". Journal of Geometry and Physics 93: 52. doi:10.1016/j.geomphys.2015.03.004. Bibcode: 2015JGP....93...52A. 
• Angella, Daniele (2015-07-25). "On the Bott-Chern and Aeppli cohomology". arXiv:1507.07112 [math.CV].

External links

• Bott-Chern cohomology at the nLab