L 2 -Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem
[Nombres de Betti L 2 et convergence des nombres de Hodge normalisés via le théorème d’annulation générique faible de Nakano]
Di Cerbo, Luca F.1 ; Lombardi, Luigi2
1 University of Florida (USA)
2 University of Milan (Italy)
Annales de l'Institut Fourier, Online first, 27 p.

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.

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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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) 
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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, Online first, 27 p.

[1] Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik Convergence of normalized Betti numbers in nonpositive curvature (2018) (https://arxiv.org/abs/1811.02520)

[2] Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo On the growth of L 2 -invariants for sequences of lattices in Lie groups, Ann. Math., Volume 185 (2017) no. 3, pp. 711-790 | DOI | MR

[3] Beauville, Arnaud Annulation du H 1 pour les fibrés en droites plats, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Mathematics), Volume 1507, Springer, 1992, pp. 1-15 | DOI | MR | Zbl

[4] Beauville, Arnaud Complex algebraic surfaces, London Mathematical Society Student Texts, 34, Cambridge University Press, 1996, x+132 pages (translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid) | DOI | MR

[5] Bondal, Alexey; Orlov, Dmitri Semiorthogonal decomposition for algebraic varieties (1995) (https://arxiv.org/abs/alg-geom/9506012)

[6] Budur, Nero Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers, Adv. Math., Volume 221 (2009) no. 1, pp. 217-250 | DOI | MR

[7] Cartwright, Donald I.; Koziarz, Vincent; Yeung, Sai-Kee On the Cartwright-Steger surface, J. Algebr. Geom., Volume 26 (2017) no. 4, pp. 655-689 | DOI | MR

[8] Cartwright, Donald I.; Steger, Tim Enumeration of the 50 fake projective planes, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 1-2, pp. 11-13 | DOI | MR

[9] DeGeorge, David L.; Wallach, Nolan R. Limit formulas for multiplicities in L 2 (Γ∖G), Ann. Math., Volume 107 (1978) no. 1, pp. 133-150 | DOI | MR

[10] DeGeorge, David L.; Wallach, Nolan R. Limit formulas for multiplicities in L 2 (Γ∖G). II. The tempered spectrum, Ann. Math., Volume 109 (1979) no. 3, pp. 477-495 | DOI | MR

[11] Di Cerbo, Luca F.; Lombardi, Luigi Moving Seshadri constants, and coverings of varieties of maximal Albanese dimension, Asian J. Math., Volume 25 (2021) no. 2, pp. 305-320 | DOI | MR

[12] Di Cerbo, Luca F.; Stern, Mark Price inequalities and Betti number growth on manifolds without conjugate points (2017) (to appear in Commun. Anal. Geom., https://arxiv.org/abs/1704.06354)

[13] Green, Mark; Lazarsfeld, Robert Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math., Volume 90 (1987) no. 2, pp. 389-407 | DOI | MR

[14] Green, Mark; Lazarsfeld, Robert Higher obstructions to deforming cohomology groups of line bundles, J. Am. Math. Soc., Volume 4 (1991) no. 1, pp. 87-103 | DOI | MR

[15] Grieve, Nathan Index conditions and cup-product maps on Abelian varieties, Int. J. Math., Volume 25 (2014) no. 4, 1450036, 31 pages | DOI | MR

[16] Gromov, Mikhael Kähler hyperbolicity and L 2 -Hodge theory, J. Differ. Geom., Volume 33 (1991) no. 1, pp. 263-292 | MR

[17] Hacon, Christopher; Pardini, Rita On the birational geometry of varieties of maximal Albanese dimension, J. Reine Angew. Math., Volume 546 (2002), pp. 177-199 | DOI | MR

[18] Hacon, Christopher; Popa, Mihnea; Schnell, Christian Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun, Local and global methods in algebraic geometry (Contemporary Mathematics), Volume 712, American Mathematical Society, 2018, pp. 143-195 | DOI | MR

[19] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | MR

[20] Jost, Jürgen; Zuo, Kang Vanishing theorems for L 2 -cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry, Commun. Anal. Geom., Volume 8 (2000) no. 1, pp. 1-30 | DOI | MR

[21] Kajdan, David A. On arithmetic varieties, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) (1975), pp. 151-217 | MR

[22] Kollár, János Shafarevich maps and plurigenera of algebraic varieties, Invent. Math., Volume 113 (1993) no. 1, pp. 177-215 | DOI | MR

[23] Lazarsfeld, Robert Positivity in algebraic geometry I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 48, Springer, 2004, xviii+387 pages | DOI | MR

[24] Lazarsfeld, Robert Positivity in algebraic geometry II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 49, Springer, 2004, xviii+385 pages | DOI | MR

[25] Liu, Yongqiang; Maxim, Laurentiu; Wang, Botong Generic vanishing for semi-abelian varieties and integral Alexander modules, Math. Z., Volume 293 (2019) no. 1-2, pp. 629-645 | DOI | MR

[26] Liu, Yongqiang; Maxim, Laurentiu; Wang, Botong Topology of subvarieties of complex semi-abelian varieties, Int. Math. Res. Not. (2021) no. 14, pp. 11169-11208 | DOI | MR

[27] Lombardi, Luigi Inequalities for the Hodge numbers of irregular compact Kähler manifolds, Int. Math. Res. Not. (2013) no. 1, pp. 63-83 | DOI | MR

[28] Lombardi, Luigi Derived invariants of irregular varieties and Hochschild homology, Algebra Number Theory, Volume 8 (2014) no. 3, pp. 513-542 | DOI | MR

[29] Lombardi, Luigi; Popa, Mihnea; Schnell, Christian Pushforwards of pluricanonical bundles under morphisms to abelian varieties, J. Eur. Math. Soc., Volume 22 (2020) no. 8, pp. 2511-2536 | DOI | MR

[30] Lück, Wolfgang Approximating L 2 -invariants by their finite-dimensional analogues, Geom. Funct. Anal., Volume 4 (1994) no. 4, pp. 455-481 | DOI | MR

[31] Lück, Wolfgang L 2 -invariants: theory and applications to geometry and K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 44, Springer, 2002, xvi+595 pages | DOI | MR

[32] Mumford, David Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Tata Institute of Fundamental Research; Hindustan Book Agency, 2008, xii+263 pages with appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition | MR

[33] Pareschi, Giuseppe; Popa, Mihnea GV-sheaves, Fourier-Mukai transform, and generic vanishing, Am. J. Math., Volume 133 (2011) no. 1, pp. 235-271 | DOI | MR

[34] Popa, Mihnea; Schnell, Christian Generic vanishing theory via mixed Hodge modules, Forum Math. Sigma, Volume 1 (2013), e1, 60 pages | DOI | MR

[35] Popa, Mihnea; Schnell, Christian Kodaira dimension and zeros of holomorphic one-forms, Ann. Math., Volume 179 (2014) no. 3, pp. 1109-1120 | DOI | MR

[36] Popa, Mihnea; Schnell, Christian On direct images of pluricanonical bundles, Algebra Number Theory, Volume 8 (2014) no. 9, pp. 2273-2295 | DOI | MR

[37] Raynaud, Michel Sous-variétés d’une variété abélienne et points de torsion, Arithmetic and geometry, Vol. I (Progress in Mathematics), Volume 35, Birkhäuser, 1983, pp. 327-352 | MR

[38] Schnell, Christian Notes on generic vanishing (2013) (available on the author’s webpage)

[39] Simpson, Carlos Subspaces of moduli spaces of rank one local systems, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 3, pp. 361-401 | MR

[40] Stover, Matthew On general type surfaces with q=1 and c 2 =3p g , Manuscr. Math., Volume 159 (2019) no. 1-2, pp. 171-182 | DOI | MR

[41] Vidussi, Stefano The slope of surfaces with Albanese dimension one, Math. Proc. Camb. Philos. Soc., Volume 167 (2019) no. 2, pp. 355-360 | DOI | MR

[42] Yeung, Sai-Kee Betti numbers on a tower of coverings, Duke Math. J., Volume 73 (1994) no. 1, pp. 201-226 | DOI | MR

[43] Zhang, Tong Severi inequality for varieties of maximal Albanese dimension, Math. Ann., Volume 359 (2014) no. 3-4, pp. 1097-1114 | DOI | MR

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