Dolbeault cohomology

The Dolbeault cohomology is a mathematical construction from the field of differential topology and complex geometry . It was named after the mathematician Pierre Dolbeault , who defined and examined it in 1953. The Dolbeault cohomology is a special cohomology theory . As an analogue of the De Rham cohomology on complex manifolds, it is also central to the Hodge theory .

Dolbeault complex

Below will be with the amount - differential forms referred. Let be a -dimensional complex manifold , an open subset and ${\displaystyle {\mathcal {A}}^{p,q}}$ ${\displaystyle (p,q)}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle U\subset M}$

${\displaystyle {\overline {\partial }}\colon {\mathcal {A}}^{p,q}(U)\to {\mathcal {A}}^{p,q+1}(U)}$

the Dolbeault cross operator . Then the sequence is called

${\displaystyle 0\longrightarrow \mathbb {C} \cong {\mathcal {A}}^{p,0}(U){\stackrel {{\overline {\partial }}_{0}}{\longrightarrow }}{\mathcal {A}}^{p,1}(U){\stackrel {{\overline {\partial }}_{1}}{\longrightarrow }}{\mathcal {A}}^{p,2}(U){\stackrel {{\overline {\partial }}_{2}}{\longrightarrow }}\ldots {\stackrel {{\overline {\partial }}_{n-1}}{\longrightarrow }}{\mathcal {A}}^{p,n}(U)\longrightarrow 0}$

${\displaystyle p}$-ter Dolbeault complex. This complex is a coquette complex , because the following applies. Since the underlying manifold is finite-dimensional, the complex breaks off after steps. In addition, the Dolbeault complex is elliptical , that is, the coquette complex of the main symbols of is exact . ${\displaystyle {\overline {\partial }}_{k+1}\circ {\overline {\partial }}_{k}=0.}$ ${\displaystyle n}$ ${\displaystyle {\overline {\partial }}}$

Dolbeault cohomology

A cohomology is obtained from this -th coquette complex in the usual way . This cohomology is called the -th Dolbeault cohomology and is noted by. The -th cohomology group of the -th Dolbeault cohomology or the -th Dolbeault group for short is thus defined as ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle H_{\overline {\partial }}^{p}(U)}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle (q,p)}$

${\displaystyle H_{\overline {\partial }}^{p,q}(U)=\mathrm {Kern} \left({\overline {\partial }}^{q}\right)/\mathrm {Bild} \left({\overline {\partial }}^{q-1}\right).}$

Just as in the De Rham cohomology, the cohomology groups are also vector spaces .

Dolbeault's theorem

Dolbeault's theorem is a complex analogue of de Rham's theorem . With which is sheaf of holomorphic -forms on the complex manifold designated. Dolbeault's theorem now states that the -th sheaf cohomology group with values ​​in the holomorphic -forms is isomorphic to the -th cohomology group of the -th Dolbeault cohomology . In mathematical terms, this means ${\displaystyle \Omega ^{p}(M)}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle H_{G}^{q}(M,\Omega ^{p}(M))}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle H_{\overline {\partial }}^{p,q}(M)}$

${\displaystyle H_{\overline {\partial }}^{p,q}(M)\cong H_{G}^{q}(M,\Omega ^{p}(M)).}$

literature

• P. Dolbeault: Sur la cohomologie des variétés analytiques complexes . In: Comptes rendus hebdomadaires des séances de l'Académie des Sciences. 236, 1953, ISSN  0001-4036 , pp. 175-277.
• Klaus Fritzsche, Hans Grauert : From Holomorphic Functions to Complex Manifolds. Springer-Verlag, New York NY 2002, ISBN 0-387-95395-7 ( Graduate Texts in Mathematics 213).