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Hodge's index theorem: The index ([[Signature|signature]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474801.png" /> of a compact [[Kähler manifold|Kähler manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474802.png" /> of complex dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474803.png" /> can be computed by the formula
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Hodge's index theorem: The index ([[signature]]) $\sigma(M)$ of a compact [[Kähler manifold]] $M$ of complex dimension $2n$ can be computed by the formula
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$$
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\sigma(M) = \sum_{p,q\,:\,p+q\,\text{even}} (-1)^{p+q} h^{p,q}
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$$
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where $h^{p,q} = \dim H^{p,q}(M)$ is the dimension of the space of harmonic forms of type $(p,q)$ on $M$ (cf. [[Harmonic form]]). This was proved by W.V.D. Hodge .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474804.png" /></td> </tr></table>
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Hodge's theorem on the decomposition of the space of smooth sections of an elliptic complex on a compact manifold into the orthogonal direct sum of subspaces of harmonic exact and co-exact sections (see [[Laplace operator]]). This was proved by W.V.D. Hodge [[#References|[2]]] for the de Rham complex
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474805.png" /> is the dimension of the space of harmonic forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474807.png" /> (cf. [[Harmonic form|Harmonic form]]). This was proved by W.V.D. Hodge .
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E^*(M) = \sum_{p\ge0} E^p(M)
 
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$$
Hodge's theorem on the decomposition of the space of smooth sections of an elliptic complex on a compact manifold into the orthogonal direct sum of subspaces of harmonic exact and co-exact sections (see [[Laplace operator|Laplace operator]]). This was proved by W.V.D. Hodge [[#References|[2]]] for the de Rham complex
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on an orientable compact Riemannian manifold $M$. In this case Hodge's theorem asserts that for any $p\ge0$ the space $H^p(M)$ of harmonic forms on $M$ is finite-dimensional and that there exists a unique operator $G : E^p(M) \rightarrow E^p(M)$ (the Green–de Rham operator) satisfying the conditions
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474808.png" /></td> </tr></table>
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G(H^p(M)) = 0 \ ;\ \ \ Gd = dg\ ;\ \ \ G \delta = \delta G
 
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$$
on an orientable compact Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h0474809.png" />. In this case Hodge's theorem asserts that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748010.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748011.png" /> of harmonic forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748012.png" /> is finite-dimensional and that there exists a unique operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748013.png" /> (the Green–de Rham operator) satisfying the conditions
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$$
 
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E^p(M) = H^p(M) \oplus d \delta GE^p(M) \oplus \delta d G E^p(M)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748014.png" /></td> </tr></table>
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$$
 
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(the Hodge decomposition). In particular, $H^p(M)$ is isomorphic to the real cohomology space $H^p(M,\mathbf{R})$ of $M$. Another important special case is the Hodge theorem for the Dolbeault complex on a compact complex manifold $M$ (see [[Differential form]]) [[#References|[3]]]. These results lead to the classical [[Hodge structure]] in the cohomology spaces of a compact Kähler manifold.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748015.png" /></td> </tr></table>
 
 
 
(the Hodge decomposition). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748016.png" /> is isomorphic to the real cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748018.png" />. Another important special case is the Hodge theorem for the Dolbeault complex on a compact complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047480/h04748019.png" /> (see [[Differential form|Differential form]]) [[#References|[3]]]. These results lead to the classical [[Hodge structure|Hodge structure]] in the cohomology spaces of a compact Kähler manifold.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, "The topological invariants of algebraic varieties" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''1''' , Amer. Math. Soc. (1952) pp. 182–192 {{MR|0046075}} {{ZBL|0048.41701}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1962) {{MR|1015714}} {{MR|0051571}} {{MR|0003947}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) {{MR|}} {{ZBL|0534.58003}} </TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, "The topological invariants of algebraic varieties" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''1''' , Amer. Math. Soc. (1952) pp. 182–192 {{MR|0046075}} {{ZBL|0048.41701}} </TD></TR>
 
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<TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1962) {{MR|1015714}} {{MR|0051571}} {{MR|0003947}} {{ZBL|}} </TD></TR>
 
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<TR><TD valign="top">[3]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR>
====Comments====
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<TR><TD valign="top">[4]</TD> <TD valign="top"> G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) {{MR|}} {{ZBL|0534.58003}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR></table>
  
 
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{{TEX|done}}
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR></table>
 

Latest revision as of 08:52, 23 November 2023

Hodge's index theorem: The index (signature) $\sigma(M)$ of a compact Kähler manifold $M$ of complex dimension $2n$ can be computed by the formula $$ \sigma(M) = \sum_{p,q\,:\,p+q\,\text{even}} (-1)^{p+q} h^{p,q} $$ where $h^{p,q} = \dim H^{p,q}(M)$ is the dimension of the space of harmonic forms of type $(p,q)$ on $M$ (cf. Harmonic form). This was proved by W.V.D. Hodge .

Hodge's theorem on the decomposition of the space of smooth sections of an elliptic complex on a compact manifold into the orthogonal direct sum of subspaces of harmonic exact and co-exact sections (see Laplace operator). This was proved by W.V.D. Hodge [2] for the de Rham complex $$ E^*(M) = \sum_{p\ge0} E^p(M) $$ on an orientable compact Riemannian manifold $M$. In this case Hodge's theorem asserts that for any $p\ge0$ the space $H^p(M)$ of harmonic forms on $M$ is finite-dimensional and that there exists a unique operator $G : E^p(M) \rightarrow E^p(M)$ (the Green–de Rham operator) satisfying the conditions $$ G(H^p(M)) = 0 \ ;\ \ \ Gd = dg\ ;\ \ \ G \delta = \delta G $$ $$ E^p(M) = H^p(M) \oplus d \delta GE^p(M) \oplus \delta d G E^p(M) $$ (the Hodge decomposition). In particular, $H^p(M)$ is isomorphic to the real cohomology space $H^p(M,\mathbf{R})$ of $M$. Another important special case is the Hodge theorem for the Dolbeault complex on a compact complex manifold $M$ (see Differential form) [3]. These results lead to the classical Hodge structure in the cohomology spaces of a compact Kähler manifold.

References

[1] W.V.D. Hodge, "The topological invariants of algebraic varieties" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 1 , Amer. Math. Soc. (1952) pp. 182–192 MR0046075 Zbl 0048.41701
[2] W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1962) MR1015714 MR0051571 MR0003947
[3] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[4] G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) Zbl 0534.58003
[a1] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004
How to Cite This Entry:
Hodge theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hodge_theorem&oldid=23855
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article