# Hodge structure

of weight (pure)

An object consisting of a lattice in the real vector space and a decomposition of the complex vector space (a Hodge decomposition). Here the condition must hold, where the bar denotes complex conjugation in . Another description of the Hodge decomposition consists in the specification of a decreasing filtration (a Hodge filtration) in such that for . Then the subspace can be recovered by the formula .

An example is the Hodge structure in the -dimensional cohomology space of a compact Kähler manifold , which was first studied by W.V.D. Hodge (see [1]). In this case the subspace can be described as the space of harmonic forms of type (cf. Harmonic form), or as the cohomology space of sheaves of holomorphic differential forms [2]. The Hodge filtration in arises from the filtration of the sheaf complex , the -dimensional hypercohomology group of which is , by subcomplexes .

A more general concept is that of a mixed Hodge structure. This is an object consisting of a lattice in , an increasing filtration (a filtration of weights) in and a decreasing filtration (a Hodge filtration) in , such that on the space , the filtrations and determine a pure Hodge structure of weight . The mixed Hodge structure in the cohomology spaces of a complex algebraic variety (not necessarily compact or smooth) is an analogue of the structure of the Galois module in the étale cohomology (cf. [3]). The Hodge structure has important applications in algebraic geometry (see Period mapping) and in the theory of singularities of smooth mappings (see [4]).

#### References

 [1] W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952) MR0051571 [2] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001 [3] P. Deligne, "Poids dans la cohomologie des variétés algébriques" R. James (ed.) , Proc. Internat. Congress Mathematicians (Vancouver, 1974) , 1 , Canad. Math. Congress (1975) pp. 79–85 MR0432648 Zbl 0334.14011 [4] A.N. Varchenko, "Asymptotic integrals and Hodge structures" J. Soviet Math. , 27 (1984) pp. 2760–2784 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 22 (1983) pp. 130–166 Zbl 0554.58002