Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE PROJECTIVE FOURFOLDS WITH ALMOST NUMERICALLY POSITIVE CANONICAL DIVISORS

  • Published : 2006.11.30
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Abstract

Let X be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor K with every very general curve is positive (K is almost numerically positive) then every very general proper subvariety of X is of general type in ';he viewpoint of geometric Kodaira dimension. We note that the converse does not hold for simple abelian varieties.

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References

  1. F. Ambro, Nef dimension of minimal models, Math. Ann. 330 (2004), no. 2, 309-322
  2. T. Fujita, Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 685-696
  3. Y. Kawamata, On the classification of noncomplete algebraic surfaces, Lecture Notes in Math. 732 (1979), 215-232 https://doi.org/10.1007/BFb0066646
  4. Y. Kawamata, Abundance theorem for minimal threefolds, Invent. Math. 108 (1992), no. 2, 229-246 https://doi.org/10.1007/BF02100604
  5. Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimal model problem, Adv. Stud. Pure Math. 10 (1987), 283-360
  6. S. Keel, K. Matsuki, and J. McKernan, Log abundance theorem for threefolds, Duke Math. J. 75 (1994), no. 1, 99-119 https://doi.org/10.1215/S0012-7094-94-07504-2
  7. S. Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987
  8. Y. Miyaoka, On the Kodaira dimension of minimal threefolds. Math. Ann. 281 (1988), no. 2, 325-332 https://doi.org/10.1007/BF01458437
  9. Y. Miyaoka and S. Mori, A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), no. 1, 65-69 https://doi.org/10.2307/1971387
  10. S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117-253 https://doi.org/10.2307/1990969
  11. S. Mori and S. Mukai, The uniruledness of the moduli space of curves of genus 11, Lecture Notes in Math. 1016 (1983), 334-353 https://doi.org/10.1007/BFb0099970
  12. V. V. Shokurov, Three-dimensional log perestroikas, Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95-202 https://doi.org/10.1070/IM1993v040n01ABEH001862
  13. V. V. Shokurov, Prelimiting flips, Proc. Steklov Inst. Math. 240 (2003), 75-213
  14. Y.-T. Siu, Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Complex Geometry (ed. I. Bauer), Springer-Verlag, Berlin, 2002, pp. 223-277
  15. H. Tsuji, Deformation invariance of plurigenera, Nagoya Math. J. 166 (2002), 117-134 https://doi.org/10.1017/S002776300000828X
  16. K. Ueno, Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Math. 439 (1975), 1-278 https://doi.org/10.1007/BFb0070571

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