An overview of recent advances in Hodge theory. (English. Russian original) Zbl 0793.14005

Several complex variables VI. Complex manifolds, Encycl. Math. Sci. 69, 39-142 (1990); Russian translation in Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 69, 48-165 (1991).

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This article can be regarded as a report on the progress in Hodge theory since 1975.
In the first section, we review Deligne’s construction of mixed Hodge structures on the cohomology of all algebraic varieties, and simplicial varieties, over \(\mathbb{C}\), for it sets the tone of much of the work that followed [cf. P. Deligne, Théorie de Hodge. I–III, Actes Congr. internat. Math. 1970, Part I, 425-430 (1971; Zbl 0219.14006), Publ. Math., Inst. Hautes Étud. Sci. 40(1971), 5-57 (1972; Zbl 0219.14007) and ibid. 44(1974), 5-77 (1975; Zbl 0237.14003)]. – Much of section 2 also contains review material, namely the theorem of W. Schmid [Invent. Math. 22, 211-319 (1973; Zbl 0278.14003)] on degeneration of Hodge structures in the abstract, i.e., in the absence of any hypothesis that the variation of Hodge structure arises from a family of smooth projective varieties. These are the nilpotent orbit theorem and the SL(2)-orbit theorem. – Section 3 is about \(L_ 2\)-cohomology. It is an integral part of the subject. There is a general relation between \(L_ 2\)-cohomology and harmonic forms, which gives a key motivation for its introduction: it was the most familiar way to establish the existence of useful Hodge decompositions.
Ironically, it was not by \(L_ 2\)-cohomology that Hodge structures for the intersection homology groups of singular projective varieties were finally produced. Instead, it came out of work in algebraic analysis (i.e., \(\mathbb{D}\)-modules), a subject that also has blossomed during the past 15 years, as we report in section 4. The starting point is the so- called Riemann-Hilbert correspondence, which asserts that taking the de Rham complex sets up an equivalence of categories between holonomic \(\mathbb{D}\)-modules with regular singularities and perverse sheaves (and likewise for their derived categories). The idea is to equip such \(\mathbb{D}\)-modules with Hodge (and weight) filtrations, so that they induce (mixed) Hodge structures on hypercohomology. – Section 5 is devoted to Deligne cohomology and to its generalization, by Beilinson, to noncompact varieties. – Several approaches have been used to put mixed Hodge structures on homotopy groups of algebraic varieties, discussed in section 6. The method of J. W. Morgan [Publ. Math., Inst. Hautes Étud. Sci. 48, 137-204 (1978; Zbl 0401.14003) and ibid. 64, 185 (1986; Zbl 0617.14013)] is based on Sullivan’s theory of minimal models for differential graded algebras. The method of R. M. Hain [\(K\)-Theory 1, 271-324 (1987; Zbl 0637.55006) and ibid. 1, No. 5, 481-497 (1987; Zbl 0657.14004)] is based on Chen’s method of iterated integrals. Alternative approaches to mixed Hodge theory on homotopy groups, due Deligne and to Navarro Aznar, are only briefly mentioned.
The notion of a variation of mixed Hodge structure is a very natural generalization of that of a variation of Hodge structure. The idea, naturally, is that a variation of mixed Hodge structure on \(X\) must at least yield a filtered local system \((\mathbb{V},W)\) on \(X\), and that the graded local systems \(\text{Gr}^ W_ k \mathbb{V}\) should be honest polarized variations of Hodge structure. In section 7, we present such a notion, that of admissible variations of mixed Hodge structure. The definition is due to Steenbrink and Zucker in the case of curves, and to Kashiwara in the general case.
Section 8 is devoted to the question of determining which local systems on a given quasi-projective manifold underlie a variation of Hodge structure. We primarily discuss very recent work of C. Simpson, which, in our opinion, promises to have profound repercussions in Hodge theory. He uses the nonlinear P.D.E. methods of differential geometry to obtain a correspondence between irreducible vector bundles on a compact Kähler manifold and stable Higgs bundles with vanishing total Chern class (see theorem 8.9). A Higgs bundle is a vector bundle \({\mathcal E}\), together with an operator-valued one-form \(\theta\) on \({\mathcal E}\) of square 0. \(\mathbb{C}^*\) acts on Higgs bundles in the obvious way, by dilating \(\theta\). The remarkable fact is that the fixed points of this \(\mathbb{C}^*\)-action correspond exactly to complex variations of Hodge structure (corollary 8.10), and real variations correspond to fixed points which give self-dual Higgs bundles (see proposition 8.12). Some striking applications are presented.


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
14-03 History of algebraic geometry
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
01A60 History of mathematics in the 20th century