L 2 -Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem
Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 423-449.

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.

Nous étudions le taux de croissance des nombres de Hodge normalisés le long d’une tour de revêtement abéliennes d’une variété projective lisse avec l’application d’Albanese semi-petite. Ces bornes sont dans certains cas optimales. De plus, nous calculons les nombres de Betti L 2 des variétés irrégulières qui satisfont le théorème d’annulation générique faible de Nakano (e.g., variétés avec l’application d’Albanese semi-petite). Enfin, nous étudions la convergence de plurigenres normalisés le long de tours de revêtement abéliennes de toute variété irrégulière. On applique ça à l’extension d’un résultat de Kollár concernant la multiplicativité des plurigenres supérieurs d’une variété projective lisse de type général, à une classe plus large de variétés. En annexe, nous étudions les variétés irrégulières pour lesquelles le premier nombre de Betti diverge le long d’une tour de revêtement abéliennes induite par la variété d’Albanese.

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DOI: 10.5802/aif.3594
Classification: 14F06, 32J25, 32L20
Keywords: $L^2$-Betti Numbers, Normalized Hodge Numbers, Irregular Varieties
Mot clés : Nombres de Betti $L^2$, nombres de Hodge normalisés, variétés irrégulières
Di Cerbo, Luca F. 1; Lombardi, Luigi 2

1 University of Florida (USA)
2 University of Milan (Italy)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Di Cerbo, Luca F.; Lombardi, Luigi. $L^2$-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem. Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 423-449. doi : 10.5802/aif.3594. https://aif.centre-mersenne.org/articles/10.5802/aif.3594/

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