Harmonic Maps of Infinite Energy and Rigidity Results for Quasiprojective Varieties

Hodge theory is a fundamental tool of Kähler geometry. It represents cohomology classes by harmonic forms, and deriving properties of these harmonic forms then in turn yields information about the cohomology. Homology or cohomology groups, however, contain only partial information about the topology of a manifold. The most important topological invariant of a manifold X is the fundamental group π1(X). Via the Albanese period mapping, π1(X) acts as a lattice on some vector space. We thus have an abelian representation of π1(X) yielding the first homology group. In order to obtain further information about the topology via π1(X), it is then natural to also study nonabelian representations of π1(X). In the same way as the Albanese period map is harmonic (this is essentially the Hodge theorem for 1-forms), for a nonabelian representation, one tries to construct an equivariant harmonic map u and to study its properties. In the case of a compact Riemannian, Kählerian, or algebraic manifold, this theory is already well developed and too extensive to be reviewed here. Many manifolds naturally arising in geometry, however, are noncompact. Here, we want to discuss the case of quasiprojective manifolds, i.e. those that can be compactified by adding a divisor at infinity. The local topology around infinity may be quite intricate, and much of it gets lost when passing to the compactification. Iitaka [I] extended the Albanese map to quasiprojective varieties. The harmonic map theory was extended to the quasiprojective case by Jost-Yau [JY2], [JY3], and Jost-Zuo [JZ1], [JZ2]. These papers, however, deal with a situation where one can construct a harmonic map of finite energy, and this requires some additional assumptions about the representation of π1(X) near infinity. In the onedimensional case, harmonic maps of possibly infinite energy have been produced and used by Simpson [S1], and in a different context by Wolf