Hodge decomposition

The Hodge theory or the set of Hodge is a central message of the Hodge theory. This theory combines the mathematical sub-areas of analysis , differential geometry and algebraic topology . The Hodge decomposition and the Hodge theory are named after the mathematician William Vallance Douglas Hodge , who developed them in the 1930s as an extension of the De Rham cohomology .

Elliptical complex

With smooth to cuts referred to in a vector bundle. Let be an oriented Riemannian manifold and a sequence of vector bundles . An elliptic complex is a sequence of first-order partial differential operators ${\displaystyle \Gamma ^{\infty }}$ ${\displaystyle (M,g)}$ ${\displaystyle (E_{i})_{i}}$ ${\displaystyle (D_{i})_{i}}$

${\displaystyle 0\longrightarrow \Gamma ^{\infty }(E_{0}){\stackrel {D_{0}}{\longrightarrow }}\Gamma ^{\infty }(E_{1}){\stackrel {D_{1}}{\longrightarrow }}\ldots {\stackrel {D_{m-1}}{\longrightarrow }}\Gamma ^{\infty }(E_{m})\longrightarrow 0,}$

so the following properties apply.

• The result is a coquette complex , that is, it applies to everyone and ${\displaystyle (\Gamma ^{\infty }(E_{i}),D_{i})}$ ${\displaystyle D_{i}\circ D_{i-1}=0}$ ${\displaystyle 1\leq i\leq m}$
• for each is the sequence of the main symbols ${\displaystyle (x,\xi )\in T^{*}M\backslash \{0\}}$

${\displaystyle 0\longrightarrow \pi (E_{0}){\stackrel {\sigma _{D_{0}}}{\longrightarrow }}\pi (E_{1}){\stackrel {\sigma _{D_{1}}}{\longrightarrow }}\ldots {\stackrel {\sigma _{D_{m-1}}}{\longrightarrow }}\pi (E_{m})\longrightarrow 0}$

exactly. The term bundle projection. ${\displaystyle \pi \colon E_{i}\to M}$

The spaces can be understood, for example, as the spaces of differential forms . ${\displaystyle \Gamma ^{\infty }(E_{i})}$

Hodge's theorem

Let now be a compact , oriented Riemannian manifold and the i-th cohomology group of the elliptic complex . Also define a (Laplace) operator ${\displaystyle M}$ ${\displaystyle H^{i}(E_{.},D_{.})}$ ${\displaystyle (\Gamma ^{\infty }(E_{i}),D_{i})}$

${\displaystyle \Delta _{i}:\Gamma ^{\infty }(E_{i})\to \Gamma ^{\infty }(E_{i}).}$

by

${\displaystyle \Delta _{i}=D_{i}^{*}\circ D_{i}+D_{i-1}\circ D_{i-1}^{*}.}$

This is an elliptic operator . Now applies:

• The -th cohomology group is for all isomorphic to the kernel of , that is ${\displaystyle i}$ ${\displaystyle H^{i}(E.,D_{.})}$ ${\displaystyle i\in \mathbb {Z} }$ ${\displaystyle \Delta _{i}}$

${\displaystyle \forall i:\ H^{i}(E.,D_{.})\cong \ker(\Delta _{i})\subset \Gamma ^{\infty }(E_{i}).}$

• The dimension of the -th cohomology group is finite for all ${\displaystyle i}$ ${\displaystyle i\in \mathbb {Z} }$

${\displaystyle \dim H^{i}(E.,D_{.})<\infty .}$

• There is an orthogonal decomposition

${\displaystyle \Gamma ^{\infty }(E_{i})=\ker(\Delta _{i})\oplus R(D_{i-1})\oplus R(D_{i}^{*}).}$

It denotes the core and the image of an operator. ${\displaystyle \ker }$ ${\displaystyle R}$

Example: De Rham cohomology

The De Rham Complex

${\displaystyle 0\to {\mathcal {A}}^{0}(M){\xrightarrow {\mathrm {d_{0}} }}{\mathcal {A}}^{1}(M){\xrightarrow {\mathrm {d_{1}} }}\ldots {\xrightarrow {\mathrm {d} _{m-1}}}{\mathcal {A}}^{m}(M)\to 0}$

is an elliptical complex. The spaces are again the spaces of the differential forms of the i-th degree and is the external derivative . The associated sequence of main symbols is the Koszul complex . The operator is the Hodge-Laplace operator . The core of this operator is called the space of harmonic differential forms , since this is defined analogously to the space of harmonic functions . According to Hodge's theorem, there is now an isomorphism between the i-th De Rham cohomology group and the space of harmonic differential forms of degree . ${\displaystyle {\mathcal {A}}^{i}}$ ${\displaystyle \mathrm {d} _{i}}$ ${\displaystyle \Delta =\mathrm {d} ^{*}\mathrm {d} +\mathrm {d} \mathrm {d} ^{*}}$ ${\displaystyle H_{\mathrm {dR} }^{i}({\mathcal {A}}(M),\mathrm {d} )}$ ${\displaystyle \ker(\Delta _{i})}$ ${\displaystyle i}$

Also are

${\displaystyle b_{i}(M):=\dim(H_{\mathrm {dR} }^{i}({\mathcal {A}}(M),\mathrm {d} ))}$

well-defined numbers, since the De Rham cohomology groups have finite dimensions. These numbers are called Betti numbers . The Hodge star operator also induces an isomorphism between the spaces and . This is the Poincaré duality and applies to the Betti numbers ${\displaystyle \star :{\mathcal {A}}^{i}(M)\to {\mathcal {A}}^{n-i}(M)}$ ${\displaystyle \ker(\Delta _{i})}$ ${\displaystyle \ker(\Delta _{n-i})}$

${\displaystyle b_{i}(M)=b_{n-i}(M).}$

literature

• Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific, Singapore et al. 2007, ISBN 978-981-270853-3 .