Von Neumann dimension, Hodge index theorem and geometric applications. (English) Zbl 1435.32027
Summary: A reformulation of the Hodge index theorem within the framework of Atiyah’s \(L^2\)-index theory is provided. More precisely, given a compact Kähler manifold \((M, h)\) of even complex dimension \(2m,\) we prove that \[\sigma (M)=\!\!\sum _{p,q=0}^{2m}\!(-1)^ph_{(2),\Gamma }^{p,q}(M) \] where \(\sigma (M)\) is the signature of \(M\) and \(h_{(2),\Gamma }^{p,q}(M)\) are the \(L^2\)-Hodge numbers of \(M\) with respect to a Galois covering having \(\Gamma\) as group of deck transformations. Likewise we also prove an \(L^2\)-version of the Frölicher index theorem. Afterwards we give some applications of these two theorems and finally we conclude this paper by collecting other properties of the \(L^2\)-Hodge numbers.
MSC:
| 32Q15 | Kähler manifolds |
| 32Q05 | Negative curvature complex manifolds |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 32J27 | Compact Kähler manifolds: generalizations, classification |
Keywords:
von Neumann dimension; Hodge index theorem; Kähler manifolds; \(L^2\)-Hodge numbers; Kähler parabolic manifolds; Euler characteristicReferences:
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