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Kodaira dimension of irregular varieties

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Abstract

Let f:X→Y be an algebraic fiber space with general fiber F. If Y is of maximal Albanese dimension, we show that κ(X)≥κ(Y)+κ(F).

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References

  1. Chen, J.A., Hacon, C.D.: On algebraic fiber spaces over varieties of maximal Albanese dimension. Duke Math. J. 111(1) (2002)

  2. Chen, J.A., Hacon, C.D.: On Ueno’s conjecture K. Math. Ann. 345(2), 287–296 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Eisenstein, E.: Generalizations of the restriction theorem for multiplier ideals. Ann. Inst. Fourier, to appear. arXiv:1001.2841v1

  4. Hacon: A derived category approach to generic vanishing. J. Reine Angew. Math. 575, 173–187 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hartshorne, R.: Algebraic Geometry. Grad. Texts Math., vol. 52. Springer, Berlin (1977)

    MATH  Google Scholar 

  6. Kawamata, Y.: Characterization of Abelian varieties. Compos. Math. 43, 253–276 (1981)

    MathSciNet  MATH  Google Scholar 

  7. Kawamata, Y.: Kodaira dimension of algebraic fiber spaces over curves. Invent. Math. 66, 57–71 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1–46 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kollár, J.: Subadditivity of the Kodaira dimension: fibers of general type. In: Algebraic Geometry, Sendai, 1985. Adv. Stud. Pure Math., vol. 10, pp. 361–398. North-Holland, Amsterdam (1987)

    Google Scholar 

  10. Kollár, J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. 113(1), 177–215 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kollár, J.: Shafarevich Maps and Automorphic Forms. M. B. Porter Lectures. Princeton Univ. Press, Princeton (1995)

    MATH  Google Scholar 

  12. Lai, C.-J.: Varieties fibered by good minimal models. Math. Ann. (2010). doi:10.1007/s00208-010-0574-7. arXiv:0912.3012v2

    Google Scholar 

  13. Lazarsfeld, R.: Positivity in Algebraic Geometry. II. Ergebnisse der Mathematik und ihrer Grenzge biete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49. Springer, Berlin (2004). Positivity for vector bundles, and multiplier ideals

    Book  MATH  Google Scholar 

  14. Mori, S.: Classification of higher-dimensional varieties. In: Algebraic Geometry, Bowdoin, Brunswick, Maine, 1985. Proc. Sympos. Pure Math., Part 1, vol. 46, pp. 269–331. Am. Math. Soc., Providence (1987)

    Google Scholar 

  15. Mukai, S.: Duality between D(X) and \(D(\hat{X})\) with its application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981)

    MathSciNet  MATH  Google Scholar 

  16. Mumford, D.: Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Oxford University Press, London (1970) Published for the Tata Institute of Fundamental Research, Bombay, pp. viii+242

    MATH  Google Scholar 

  17. Viehweg, E.: Die additivität der Kodaira Dimension für projektive Fasseraüme über Varietäten des allgemeinen Typs. J. Reine Angew. Math. 330, 132–142 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Viehweg, E.: Weak positivity and the additivity of the Kodaira dimension of certain fiber spaces. Adv. Stud. Pure Math. 1, 329–353 (1983)

    MathSciNet  Google Scholar 

  19. Viehweg, E.: Quasi-projectivity Moduli for Polarized Manifolds. Ergebn. Math., vol. 30. Springer, Berlin (1995)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan

    Jungkai Alfred Chen

  2. Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT, 84112, USA

    Christopher D. Hacon

Authors
  1. Jungkai Alfred Chen
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  2. Christopher D. Hacon
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Corresponding author

Correspondence to Christopher D. Hacon.

Additional information

We are in debt to O. Debarre, Z. Jiang, R. Lazarsfeld, J. Kollár, M. Popa and the referee for many useful comments and suggestions. The first author was partially supported by TIMS, NCTS/TPE and National Science Council of Taiwan. The second author was partially supported by NSF Grant 0757897.

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Chen, J.A., Hacon, C.D. Kodaira dimension of irregular varieties. Invent. math. 186, 481–500 (2011). https://doi.org/10.1007/s00222-011-0323-x

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