In this nicely written survey article, the author gives a detailed account of her work on Hodge theory of compact Kähler manifolds with a particular emphasis on the case of complex projective varieties. The special features of Hodge theory on algebraic varieties are investigated along two guidelines: the existence of a gap between the topology of complex projective varieties and general compact Kähler manifolds, and the notion of absolute Hodge classes (and Hodge loci) in the algebraic case.
The first sections of the paper are thus devoted to illustrate this gap through the examples she discovered in [Invent. Math. 157, No. 2, 329–343 (2004; Zbl 1065.32010)] and [J. Differ. Geom. 72, No. 1, 43–71 (2006; Zbl 1102.32008)] and which give a negative answer to the Kodaira problem: there exist (in dimension at least 4) compact Kähler manifolds which do not have the homotopy type of projective varieties. Actually, the cohomology algebra of the examples above (which are obtained as blow-ups of certain non-projective tori) cannot be isomorphic to the cohomology algebra of a projective manifold. The interaction between the rational polarization and the product structure on the cohomology algebra of a projective manifold gives then rise to new constraints in the algebraic case. One of the main tools used in this study is a lemma due to Deligne which provides a criterion to detect Hodge substructure in cohomology algebras. It is quite surprising that this approach also leads to the construction of new examples of symplectic manifolds which satisfy the classical constraints of Hodge theory (such as the hard Lefschetz theorem) but not having the cohomology algebra of a compact Kähler manifold. The construction is reasonably simple since these symplectic manifolds are realized as projective bundles over complex tori.
The Hodge classes are the main subject of the last section and (pursuing the first guideline) the author establishes first that their behaviour is very different in the general compact Kähler case than in the algebraic one. There indeed exists a compact Kähler manifold \(X\) (a general Weil torus for instance) which admits Hodge classes of bidegree (2,2) but with the property that \(c_2(\mathcal{F})=0\) for any coherent sheaf \(\mathcal{F}\) on \(X\). In particular, the Hodge Conjecture has no natural generalization in the compact Kähler setting.
The remaining part of the paper focuses on the Hodge classes in the projective case. The fact that the cohomology of a projective variety can be computed algebraically (through the GAGA principles and the algebraic de Rham complex) enables to define the notion of an absolute Hodge class. These are the classes which remain rational on any conjugate of the variety (under the group of field automorphisms of \(\mathbb{C}\)). The cycle classes are easily seen to be absolute, and the Hodge conjecture has the following implication: any Hodge class is absolute.
The last results of the paper are related to the algebraicity of Hodge loci of a given Hodge class \(\alpha\) on a projective variety \(X\) (shown in [E. Cattani, P. Deligne and A. Kaplan, J. Am. Math. Soc. 8, No. 2, 483–506 (1995; Zbl 0851.14004)]); these are (roughly) the points in the full family of deformations of \(X\) where the class \(\alpha\) remains a Hodge class. Some questions concerning arithmetical aspects of Hodge conjecture are finally discussed and connected to the fields of definition of the Hodge loci.