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Geography of Irregular Gorenstein 3–folds

Published online by Cambridge University Press:  20 November 2018

Tong Zhang*
Department of Mathematics, University of Alberta, Edmonton, AB T6G 2G1 e-mail:
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In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether–Castelnuovo type inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.

Research Article
Copyright © Canadian Mathematical Society 2015


[1] Barja, M. A., Lower bounds of the slope of fibred threefolds. Internat. J. Math. 11(2000), no. 4, 461491.Google Scholar
[2] Barja, M. A., Generalized Clifford–Severi inequality and the volume of irregular varieties. arxiv:1303.3045v2.Google Scholar
[3] Beauville, A., L'application canonique pour les surfaces de type général. Invent. Math. 55(1979), no. 2, 121140. http://dx.doi.Org/10.1007/BF01390086 Google Scholar
[4] Birkenhake, C. and Lange, H., Complex abelian varieties. Grundlehren der Mathematischen Wissenschaften, 302, Springer–Verlag, Berlin, 1992.Google Scholar
[5] Bombieri, E., Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. 42(1973), 171219.Google Scholar
[6] Catanese, F., Chen, M., and –Q. Zhang, D., The Noether inequality for smooth minimal 3–folds. Math. Res. Lett. 13(2006), no. 4, 653666. Google Scholar
[7] Chen, J. A. and Chen, M., The Noether inequality for Gorenstein minimal 3–folds. arxiv:1310.7709Google Scholar
[8] Chen, J. A. and Chen, M., Explicit birational geometry of threefolds of general type, I. Ann. Sci. Éc Norm. Super. 43(2010), 365394.Google Scholar
[9] Chen, J. A., Chen, M., and –Q. Zhang, D., On the 5–canonical system of 3–folds of general type. J Reine Angew. Math. 603(2007), 161181.Google Scholar
[10] Chen, J. A. and Hacon, C. D., On the geography of threefolds of general type. J. Algebra 321(2009), no. 9, 25002507.http://dx.doi.Org/10.1016/j.jalgebra.2009.01.01 2 Google Scholar
[11] Chen, M., Inequalities of Noether type for 3–folds of general type. J. Math. Soc. Japan 56(2004), no. 4, 11311155. Google Scholar
[12] Chen, M., Minimal threefolds of small slope and the Noether inequality for canonically polarized threefolds. Math. Res. Lett. 11(2004), no. 5–6, 833852. Google Scholar
[13] Chen, M. and Hacon, C. D., On the geography of Gorenstein minimal 3–folds of general type. Asian J. Math. 10(2006), no. 4, 757763. Google Scholar
[14] Debarre, O., Inégalités numériques pour les surfaces de type général. Bull. Soc. Math. France 110(1982), no. 3, 319346.Google Scholar
[15] di Severi, F., La série canonica e la teoria délie série principali degruppi dipunti sopra una superifcie algebrica. Comment. Math. Helv. 4(1932), no. 1, 268326. Google Scholar
[16] Ein, L. and Lazarsfeld, R., Singularities oftheta divisors and the birational geometry of irregular varieties. J. Amer. Math. Soc. 10(1997), no. 1, 243258. Google Scholar
[17] Francia, P., On the base points of the bicanonical system. In: Problems in the theory of surfaces and their classification (Cortona, 1988), Sympos. Math., 32, Academic Press, London, 1991, pp. 141150.Google Scholar
[18] Green, M. and Lazarsfeld, R., Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville. Invent. Math. 90(1987), no. 2, 389407.http://dx.doi.Org/10.1007/BF01388711 Google Scholar
[19] Hacon, C. and Kovâcs, S., Generic vanishing fails for singular varieties and in characteristic p > O. arxiv:1212.5105+O.+arxiv:1212.5105>Google Scholar
[20] Horikawa, E., Algebraic surfaces of general type with small c2 1 V. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(1981), no. 3, 745755.Google Scholar
[21] Hu, Y., Inequality of Noether type for smooth minimal 3-folds of general type. arxiv:1309.4618Google Scholar
[22] Hunt, B., Complex manifold geography in dimension 2 and 3. J. Differential Geom. 30(1989), no. 1, 51153.Google Scholar
[23] Kollâr, J., Higher direct images of dualizing sheaves. I. Ann. of Math. (2) 123(1986), no. 1,1142. http://dx.doi.Org/10.2307/1971351 Google Scholar
[24] Kollâr, J., Higher direct images of dualizing sheaves. II. Ann. of Math. (2) 124(1986), no. 1, 171202. http://dx.doi.Org/10.2307/1971390 Google Scholar
[25] Liedtke, C., Algebraic surfaces in positive characteristic. In: Birational geometry, rational curves, and arithmetic. Springer, New York, 2013, pp. 229292. http://dx.doi.Org/10.1007/978-1-4614-6482-2.11 Google Scholar
[26] Liedtke, C., Algebraic surfaces of general type with small c2 1 in positive characteristic. Nagoya Math. J. 191(2008), 111134.Google Scholar
[27] Lu, S. S. Y., On surfaces of general type with maximal Albanese dimension. J. Reine Angew. Math. 641(2010), 163175.Google Scholar
[28] Lu, S. S. Y., Holomorphic curves on irregular varieties of general type starting from surfaces. In: Affine algebraic geometry, CRM Proc. Lecture Notes, 54, American Mathematical Society, Providence, RI, 2011, pp. 205220.Google Scholar
[29] Lu, S. S. Y., A uniform bound on the canonical degree of Albanese defective curves on surfaces. Bull. Lond. Math. Soc. 44(2012), no. 6, 11821188.http://dx.doi.Org/10.1112/blms/bdsO43 Google Scholar
[30] Miyaoka, Y. The Chern classes and Kodaira dimension of a minimal variety. In: Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, pp. 449476.Google Scholar
[31] Ohno, K., Some inequalities for minimal fibrations of surfaces of general type over curves. J. Math. Soc. Japan 44(1992), no. 4, 643666. http://dx.doi.Org/10.2969/jmsj704440643 Google Scholar
[32] Pardini, R., The Severi inequality K2 > 4Xfor surfaces of maximal Albanese dimension. Invent. Math. 159(2005), no. 3, 669672. +4Xfor+surfaces+of+maximal+Albanese+dimension.+Invent.+Math.+159(2005),+no.+3,+669–672.+>Google Scholar
[33] Shin, D.-K., Noether inequality for a nef and big divisor on a surface. Commun. Korean Math. Soc. 23(2008), no. 1, 1118.http://dx.doi.Org/10.4134/CKMS.2008.23.1.011 Google Scholar
[34] Xiao, G., Fibered algebraic surfaces with low slope. Math. Ann. 276(1987), no. 3, 449466.http://dx.doi.Org/10.1007/BF01450841 Google Scholar
[35] Yuan, X. and Zhang, T., Relative Noether inequality on fibered surfaces. Adv. Math. 259(2014), 89115.http://dx.doi.Org/10.1016/j.aim.2014.03.018 Google Scholar
[36] Zhang, T., Severi inequality for varieties of maximal Albanese dimension. Math. Ann. 359(2014),no. 3-4, 10971114. Google Scholar