Betti numbers on a tower of coverings

Yeung, Sai Kee

doi:10.1215/s0012-7094-94-07308-0

Cited by 22 publications

(13 citation statements)

References 19 publications

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“…The paper of Donnelly [75] deals with the operator f (∆ 0 ) acting on 0-forms for a function f ∈ C ∞ 0 (R + ). Theorem 5.1 is proven by Yeung [254] in the special case of a closed Kähler manifold with negative sectional curvature and by DeGeorge-Wallach [102], [103] in the special case of a closed locally symmetric space of non-compact type. In the last case all the L 2 -Betti numbers vanish, so one gets…”

Section: -Betti Numbers By Ordinary Betti Numbersmentioning

confidence: 99%

L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

Lück

2001

Handbook of Geometric Topology

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Section: -Betti Numbers By Ordinary Betti Numbersmentioning

confidence: 99%

L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

Lück

2001

Handbook of Geometric Topology

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“…For a tower of coverings on a compact complex manifold, we exhibit a connection between Bergman stability and the theory of L 2 -Betti numbers, an area studied extensively in the literatures (cf. [11,12,33,55]; see Section 4 below).…”

Section: Introductionmentioning

confidence: 99%

Stability of the Bergman kernel on a tower of coverings

Chen¹,

Fu²

2016

J. Differential Geom.

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We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective version of Rhodes' result is given for a tower of coverings on a compact Riemann surface of genus ≥ 2. Stability of the Bergman kernel is established for towers of coverings on hyperbolic Riemann surfaces and on complete Kähler manifolds satisfying certain potential conditions. As a consequence, stability of the Bergman kernel is established for any tower of coverings of Riemann surfaces when the top manifold is simply-connected. (2000): 32Q26, 32A25, 32W05, 32J25. Mathematics Subject Classification

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“…The techniques employed both in [DW78,DW79] and [ABBGNRS17] are based on representation theory, and they do not immediately generalize to non-symmetric varieties. Nevertheless, there is a large and growing literature concerning these kind of problems outside the locally symmetric context; see for example [Yeu94], [DS17], [ABBG18] and the bibliography therein. These papers employ geometric analysis techniques, and they extend much of the DeGeorge-Wallach theory to negatively curved compact Riemannian manifolds which are not locally symmetric.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

Cerbo¹,

Lombardi²

2019

Preprint

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We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the L 2 -Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem (e.g., varieties with semismall Albanese map). Finally, we study the convergence of normalized plurigenera along towers of abelian covers of any irregular variety. As an application, we extend a result of Kollár concerning the multiplicativity of higher plurigenera of a smooth projective variety of general type, to a wider class of varieties. In the Appendix, we study irregular varieties for which the first Betti number diverges along a tower of abelian covers induced by the Albanese variety.

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Betti numbers on a tower of coverings

Cited by 22 publications

References 19 publications

L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

Stability of the Bergman kernel on a tower of coverings

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

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