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On the \(L^2\)-\(\overline{\partial }\)-cohomology of certain complete Kähler metrics

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Abstract

Let V be a compact and irreducible complex space of complex dimension v whose regular part is endowed with a complete Hermitian metric h. Let \(\pi :M\rightarrow V\) be a resolution of V. Under suitable assumptions on h we prove that

$$\begin{aligned} H^{v,q}_{2,\overline{\partial }}( \mathrm {reg} (V),h)\cong H^{v,q}_{\overline{\partial }}(M),\quad q=0,\ldots ,v. \end{aligned}$$

Then we show that the previous isomorphism applies to the case of Saper-type Kähler metrics, as introduced by Grant Melles and Milman, and to the case of complete Kähler metrics with finite volume and pinched negative sectional curvatures.

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Acknowledgements

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). Moreover the first author wishes to thank also SFB 647: Raum-Zeit-Materie for financial support. Part of this work was done while the first author was visiting Sapienza Università di Roma whose hospitality and financial support are gratefully acknowledged. It is a pleasure to thank Pierre Albin for interesting discussions. We also wish to thank the referee of a first version of this paper for very interesting remarks and suggestions.

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  1. Institut Camille Jordan, Université Lyon 1, Villeurbanne, France

    Francesco Bei

  2. Dipartimento di Matematica, Sapienza Università di Roma, Rome, Italy

    Paolo Piazza

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  1. Francesco Bei
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  2. Paolo Piazza
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Correspondence to Paolo Piazza.

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Bei, F., Piazza, P. On the \(L^2\)-\(\overline{\partial }\)-cohomology of certain complete Kähler metrics. Math. Z. 290, 521–537 (2018). https://doi.org/10.1007/s00209-017-2029-2

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