complex geometry In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Bott–Chern cohomology is a

cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

for

complex manifolds In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas of charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are holom ...

. It serves as a bridge between

de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...

, 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 similiar 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 Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott function ...

and Shiing-Chen Chern, who introduced it in 1965.

Definition

For a complex manifold

X

, its ''Bott–Chern cohomology'' is given by:Bott & Chern 1965, p. 74Angella & Tomassini 2014, p. 1 & 1.1. Bott-Chern cohomologyAngella 2015, p. 5 :

H_\mathrm^(X)
:=\ker(\mathrm_)/\operatorname(\partial_\overline\partial_)
=\left(\ker(\partial_)\cap\ker(\overline\partial_)\right)/\operatorname(\partial_\overline\partial_).

\mathrm

denotes the

exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...

while

\partial

and

\overline\partial

denote the Dobeault operators.

Maps

de Rham and Dobeault cohomology are given by:Angella 2015, p. 3-4 :

H_\mathrm^n(X)
:=\ker(\mathrm_n)/\operatorname(\mathrm_),

H_\partial^(X)
:=\ker(\partial_)/\operatorname(\partial_),

H_\overline\partial^(X)
:=\ker(\overline\partial_)/\operatorname(\overline\partial_).

Since there is a canonical inclusion

\operatorname(\partial_\overline\partial_)
=\operatorname(\overline\partial_\partial_)\hookrightarrow\operatorname(\mathrm_)

, there is a canonical map from Bott–Chern cohomology into de Rham cohomology: :

H_\mathrm^(X)\rightarrow H_\mathrm^(X).

Since there are canonical inclusions

\ker(\partial_)\cap\ker(\overline\partial_)\hookrightarrow\ker(\partial_),\ker(\overline\partial_)

as well as

\operatorname(\partial_\overline\partial_)\hookrightarrow\operatorname(\partial_)

and

\operatorname(\overline\partial_\partial_)\hookrightarrow\operatorname(\overline\partial_)

, there are canonical maps from Bott–Chern into Dobeault cohomology: :

H_\mathrm^(X)\rightarrow H_\partial^(X),

H_\mathrm^(X)\rightarrow H_\overline\partial^(X).

Furthermore there are canonical maps

H_\mathrm^n(X),H_\partial^(X),H_^(X)\rightarrow H_\mathrm^(X)

into Aeppli cohomology, with all three compositions

H_\mathrm^(X)\rightarrow H_\mathrm^(X)

being identical.

Literature

* * * {{cite arXiv , eprint=1507.07112 , class=math.CV , first=Daniele , last=Angella , title=On the Bott-Chern and Aeppli cohomology , date=2015-07-25

References

External links

* Bott-Chern cohomology at the ''n''Lab Complex geometry Cohomology theories