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Von Neumann dimension, Hodge index theorem and geometric applications

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Abstract

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

$$\begin{aligned} \sigma (M)=\!\!\sum _{p,q=0}^{2m}\!(-1)^ph_{(2),\Gamma }^{p,q}(M) \end{aligned}$$

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, see (3). 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.

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Notes

  1. Kähler non-elliptic in [9].

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Authors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Francesco Bei

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  1. Francesco Bei
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Correspondence to Francesco Bei.

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This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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Bei, F. Von Neumann dimension, Hodge index theorem and geometric applications. European Journal of Mathematics 5, 1212–1233 (2019). https://doi.org/10.1007/s40879-018-0269-2

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  • DOI: https://doi.org/10.1007/s40879-018-0269-2

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