Kähler hyperbolicity and \(L_ 2\)-Hodge theory. (English) Zbl 0719.53042

Gromov, M.

J. Differ. Geom. 33, No. 1, 263-292 (1991).

The main purpose of this paper is to study some bounded and d(bounded) forms to develop a generalization of the classical Hodge theory. By passing to the universal covering \(\tilde X\) of a Kähler hyperbolic manifold X and by showing that the \(L_ 2\)-Hodge number \(h^{p,q}L_ 2(\tilde X)\) vanishes if and only if \(p+q<n\), \(n=\dim X\) the author proves that the Euler characteristic \(\chi_ p(X)=\chi (\Omega^ p)\) does not vanish and \(sign \chi_ p=(-1)^{n-p}\) \((\Omega^ p=\Omega^ p(X)\) is the sheaf of holomorphic p-forms on X). It is proved that the \(L_ 2\)-space \(L_ 2\Omega^ p\) of exterior p-forms on a complete manifold X admits Hodge decomposition. The Lefschetz vanishing theorem in the noncompact case, where the dimension of the space \({\mathcal H}^ p\) of harmonic forms is infinite is also one of the problems which the author studies. Using that theorem and some other results he proves \({\mathcal H}^{p,q}=0\) for \(p+q\neq m=\dim_{{\mathbb{C}}}X\) and \({\mathcal H}^{p,q}\neq 0\) for \(p+q=m\) if X is a complete simply connected Kähler manifold whose Kähler form \(\omega\) is d(bounded), \(\Gamma\) a discrete group of isometries of X, such that X/\(\Gamma\) is compact (\({\mathcal H}^{p,q}\) is the space of harmonic \(L_ 2\) forms on X of bidegree (p,q)). Some consequences of this theorem and a lot of interesting examples are considered, too.

Reviewer: N.Bokan (Beograd)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
58A14	Hodge theory in global analysis

Keywords:

Lefschetz theorem; index theorem; Kähler hyperbolic manifold; Hodge number; Euler characteristic; Hodge decomposition; harmonic forms