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)
[2]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1 , Wiley (Interscience) (1978)
[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
[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

See also the references to Period mapping.

Comments

A Hodge structure of weight thus consists of i) a finitely-generated Abelian group ; and ii) a finite decreasing filtration on such that as soon as . A polarization of a Hodge structure of weight is a -symmetric -valued bilinear form on such that for , and such that for . The Hodge structures arising in algebraic geometry are always polarizable.

There exist classifying spaces for polarized Hodge structures with given numerical data [a1], as well as for mixed Hodge structures with a polarization on graded quotients for the weight filtration [a2]. Mixed Hodge structures form an Abelian category in which every morphism is strictly compatible with both Hodge and weight filtrations. Pure polarized Hodge structures form a Tannakian category [a3]. There exist canonical and functorial mixed Hodge structures on (local) cohomology groups of algebraic varieties , rational homotopy groups [a5], vanishing cycle groups of function germs [a6], [a7], and on the intersection homology groups of algebraic varieties with coefficients in a polarizable variation of Hodge structure [a8], [a9]. In the latter case, there is even a pure Hodge structure. At this moment (1989), the ultimate generalization seems to be the concept of a mixed Hodge module [a10]–[a11].

References

[a1]	P. Griffiths, "Periods of integrals on algebraic manifolds" Amer. J. Math. , 90 (1968) pp. 568–625; 805–865
[a2]	J. Carlson, E. Cattani, A. Kaplan, "Mixed Hodge structures and compactifications of Siegel's space" A. Beauville (ed.) , Algebraic geometry (Angers, 1979) , Sijthoff & Noordhoff (1980) pp. 77–105
[a3]	N. Saavedra Rivano, "Catégories Tannakiennes" , Lect. notes in math. , 265 , Springer (1972)
[a4a]	P. Deligne, "Théorie de Hodge II, III" Publ. Math. IHES , 40 (1971) pp. 5–58
[a4b]	P. Deligne, "Théorie de Hodge IV" Publ. Math. IHES , 44 (1975) pp. 5–77
[a5]	R.M. Hain, "The de Rham homotopy theory of complex algebraic varieties I, II" -theory , 1 (1987) pp. 271–324; 481–497
[a6]	J.H.M. Steenbrink, "Mixed Hodge structure on the vanishing cohomology" P. Holm (ed.) , Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School , Sijthoff & Noordhoff (1977) pp. 525–563
[a7]	V. Navarro Aznar, "Sur la théorie de Hodge–Deligne" Invent. Math. , 90 (1987) pp. 11–76
[a8]	E. Cattani, A. Kaplan, W. Schmid, " and intersection cohomologies for a polarizable variation of Hodge structure" Invent. Math. , 87 (1987) pp. 217–252
[a9]	M. Kashiwara, T. Kawai, "The Poincaré lemma for variations of Hodge structure" Publ. RIMS Kyoto Univ. , 23 : 2 (1987) pp. 345–407
[a10]	M. Saito, "Modules de Hodge polarisables" Preprint RIMS , 553 (Oct. 1986)
[a11]	M. Saito, "Mixed Hodge modules" Preprint RIMS , 585 (July 1987)