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