# zbMATH — the first resource for mathematics

Generalized Clifford-Severi inequality and the volume of irregular varieties. (English) Zbl 1409.14013
Summary: We give a sharp lower bound for the self-intersection of a nef line bundle $$L$$ on an irregular variety $$X$$ in terms of its continuous global sections and the Albanese dimension of $$X$$, which we call the generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when $$X$$ is of maximal Albanese dimension the bound is $$\mathrm{vol}(X)\geq2n!\chi(\omega_X)$$ and it is sharp.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14J10 Families, moduli, classification: algebraic theory 14J29 Surfaces of general type 14J30 $$3$$-folds 14J35 $$4$$-folds
Full Text:
##### References:
 [1] M. A. Barja, M. Lahoz, J. C. Naranjo, and G. Pareschi, On the bicanonical map of irregular varieties , J. Algebraic Geom. 21 (2012), 445-471. · Zbl 1245.14007 [2] M. A. Barja and L. Stoppino, “Stability conditions and positivity of invariants of fibrations” in Algebraic and Complex Geometry , Springer Proc. Math. Stat. 71 , Springer, Heidelberg, 2014, 1-40. · Zbl 1312.14050 [3] M. C. Beltrametti and A. J. Sommese, The Adjuntion Theory of Complex Projective Varieties , de Gruyter Exp. Math. 16 (1991), de Gruyter, Berlin. [4] F. Catanese, “Moduli of surfaces of general type” in Algebraic Geometry-Open Problems (Ravello, 1982) , Lecture Notes in Math. 997 , Springer, Berlin, 1983, 90-112. [5] F. Catanese, C. Ciliberto, and M. Mendes Lopes, On the classification of irregular surfaces of general type with nonbirational bicanonical map , Trans. Amer. Math. Soc. 350 , no. 1(1998), 275-308. · Zbl 0889.14019 [6] J. A. Chen and M. Chen, The canonical volume of 3-folds of general type with $$\chi\leq0$$ , J. Lond. Math. Soc. (2) 78 (2008), 693-706. · Zbl 1156.14009 [7] M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type , Math. Ann. 337 (2007), 887-908. · Zbl 1124.14038 [8] C. Ciliberto, M. Mendes Lopes, and R. Pardini, The classification of minimal irregular surfaces of general type with $$K^{2}=2p_{g}$$ , Algebraic Geom. 1 (2014), 479-488. · Zbl 1322.14059 [9] M. Cornalba and J. Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves , Ann. Sci. Éc. Norm. Supér. (4) 21 (1988), 455-475. · Zbl 0674.14006 [10] O. Debarre, On coverings of simple abelian varieties , Bull. Soc. Math. France 134 (2006), 253-260. · Zbl 1109.14017 [11] T. Fujita, On Kahler fiber spaces over curves , J. Math. Soc. Japan 30 (1978), 779-794. · Zbl 0393.14006 [12] C. D. Hacon and R. Pardini, Birational characterization of products of curves of genus 2 , Math. Res. Lett. 12 (2005), 129-140. · Zbl 1070.14043 [13] K. Konno, A lower bound of the slope of trigonal fibrations , Internat. J. Math. 7 (1996), 19-27. · Zbl 0880.14009 [14] H. Lange and C. Birkenhake, Complex Abelian Varieties , Grundlehren Math. Wiss. 302 , Springer, Berlin, 1992. · Zbl 0779.14012 [15] M. Manetti, Surfaces of Albanese general type and the Severi conjecture , Math. Nachr. 261/262 (2003), 105-122. · Zbl 1044.14017 [16] M. Mendes Lopes and R. Pardini, Severi type inequalities for irregular surfaces with ample canonical class , Comment. Math. Helv. 86 (2011), 401-414. · Zbl 1210.14040 [17] M. Mendes Lopes and R. Pardini, “The geography of irregular surfaces” in Current Developments in Algebraic Geometry , Math. Sci. Res. Inst. Publ. 59 , Cambridge Univ. Press, Cambridge, 2012, 349-378. · Zbl 1255.14028 [18] M. Mendes Lopes, R. Pardini, and G. P. Pirola, Continuous families of divisors, paracanonical systems and a new inequality for varieties of maximal Albanese dimension , Geom. Topol. 17 (2013), 1205-1223. · Zbl 1316.14016 [19] M. Mendes Lopes, R. Pardini, and G. P. Pirola, Brill-Noether loci for divisors on irregular varieties , J. Eur. Math. Soc. (JEMS) 16 (2014), 2033-2057. · Zbl 1317.14019 [20] D. Mumford, On the equations defining abelian varieties, I , Invent. Math. 1 (1966), 287-354. · Zbl 0219.14024 [21] D. Mumford, “Varieties defined by quadratic equations” in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) , Edizioni Cremonese, Rome, 1969, 29-100. [22] K. Ohno, Some inequalities for minimal fibrations of surfaces of general type over curves , J. Math. Soc. Japan 44 (1992), 643-666. · Zbl 0783.14021 [23] R. Pardini, The Severi inequality $$K^{2}\geq4\chi(\omega_{S})$$ for surfaces of maximal Albanese dimension , Invent. Math. 159 (2005), 669-672. · Zbl 1082.14041 [24] G. Pareschi, “Basic results on irregular varieties via Fourier-Mukai methods” in Current Developments in Algebraic Geometry , Math. Sci. Res. Inst. Publ. 59 , Cambridge Univ. Press, Cambridge, 2012, 379-403. · Zbl 1256.14016 [25] G. Pareschi and M. Popa, Regularity on abelian varieties, I , J. Amer. Math. Soc. 16 (2003), 285-302. · Zbl 1022.14012 [26] G. Pareschi and M. Popa, GV-sheaves, Fourier-Mukai transform, and generic vanishing , Amer. J. Math. 133 (2011), 235-271. · Zbl 1208.14015 [27] G. P. Pirola, personal communication, 2013. [28] F. Severi, La serie canonica e la teoria delle serie principali di gruppi di punti sopra una superficie algebrica , Comment. Math. Helv. 4 (1932), 268-326. · Zbl 0005.17602 [29] G. Xiao, Fibered algebraic surfaces with low slope , Math. Ann. 276 (1987), 449-466. · Zbl 0596.14028 [30] L. Zhang, An orbifold approach to Severi inequality , preprint, [math.AG]. arXiv:1202.2656v1 [31] T. Zhang, Severi inequality for varieties of maximal Albanese dimension , Math. Ann. 359 (2014), 1097-1114. · Zbl 1307.14064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.