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Asymptotic purity for very general hypersurfaces of ℙn × ℙn of bidegree (k, k)

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Central European Journal of Mathematics

Abstract

For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on hypersurfaces of ℙn × ℙn. In particular, we show that very general hypersurfaces of bidegree (k, k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.

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References

  1. Andreotti A., Grauert H., Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 1962, 90, 193–259

    MathSciNet  MATH  Google Scholar 

  2. Bogomolov F., personal communication

  3. Bogomolov F., De Oliveira B., Hyperbolicity of nodal hypersurfaces. J. Reine Angew. Math., 2006, 596, 89–101

    Article  MathSciNet  MATH  Google Scholar 

  4. Bott R., Homogeneous vector bundles, Ann. of Math., 1957, 66(2), 203–248

    Article  MathSciNet  MATH  Google Scholar 

  5. Bouche T., Two vanishing theorems for holomorphic vector bundles of mixed sign, Math. Z., 1995, 218(4), 519–526

    Article  MathSciNet  MATH  Google Scholar 

  6. Boucksom S., On the volume of a line bundle. Internat. J. Math., 2002, 13(10), 1043–1063

    Article  MathSciNet  MATH  Google Scholar 

  7. Burr M.A., Asymptotic Cohomological Vanishing Theorems and Applications of Real Algebraic Geometry to Computer Science, PhD thesis, Courant Institute of Mathematical Sciences, New York University, 2010

  8. Cutkosky S.D., Zariski decomposition of divisors on algebraic varieties. Duke Math. J., 1986, 53(1), 149–156

    Article  MathSciNet  MATH  Google Scholar 

  9. Cutkosky S.D., Srinivas V., Periodicity of the fixed locus of multiples of a divisor on a surface. Duke Math. J., 1993, 72(3), 641–647

    Article  MathSciNet  MATH  Google Scholar 

  10. Debarre O., Ein L., Lazarsfeld R., Voisin C., Pseudoeffective and nef classes on Abelian varieties, Compos. Math., 2011, 147(6), 1793–1818

    Article  MATH  Google Scholar 

  11. Demailly J.-P., Champs magnétiques et inégalités de Morse pour la d″-cohomologie, Ann. Inst. Fourier (Grenoble), 1985, 35(4), 189–229

    Article  MathSciNet  MATH  Google Scholar 

  12. Demailly J.-P., Holomorphic morse inequalities, In: Several Complex Variables and Complex Geometry. II, Santa Cruz, July 10–30, 1989, Proc. Sympos. Pure Math., 52, American Mathematical Society, Providence, 1991, 93–114

    Google Scholar 

  13. Demailly J.-P., Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann, Milan J. Math., 2010, 78(1), 265–277

    Article  MathSciNet  MATH  Google Scholar 

  14. Demailly J.-P., A converse to the Andreotti-Grauert theorem, preprint available at http://arxiv.org/abs/1011.3635

  15. Demailly J.-P., personal communication, 2011

  16. Ein L., Lazarsfeld R., Mustaţă M., Nakamaye M., Popa M., Asymptotic invariants of line bundles, Pure Appl. Math. Q., 2005, 1(2), Special Issue: In memory of Armand Borel. I, 379–403

    MathSciNet  MATH  Google Scholar 

  17. de Fernex T., Küronya A., Lazarsfeld R., Higher cohomology of divisors on a projective variety, Math. Ann., 2007, 337(2), 443–455

    Article  MathSciNet  MATH  Google Scholar 

  18. Fujita T., Approximating Zariski decomposition of big line bundles, Kodai Math. J., 1994, 17(1), 1–3

    Article  MathSciNet  MATH  Google Scholar 

  19. Fulton W., Harris J., Representation Theory, Grad. Texts in Math., 129, Springer, New York, 1991

    Book  MATH  Google Scholar 

  20. Hacon C., McKernan J., Xu C., On the birational automorphisms of varieties of general type, preprint available at http://arxiv.org/abs/1011.1464

  21. Hering M., Küronya A., Payne S., Asymptotic cohomological functions of toric divisors. Adv. Math., 2006, 207(2), 634–645

    Article  MathSciNet  MATH  Google Scholar 

  22. Iskovskikh V.A., Vanishing theorems, J. Math. Sci. (New York), 2001, 106(5), 3258–3268

    Article  MathSciNet  MATH  Google Scholar 

  23. Kollár J., Vanishing theorems for cohomology groups. In: Algebraic Geometry, Bowdoin, July 8–26, 1985, Proc. Sympos. Pure Math., 46(1), American Mathematical Society, Providence, 1987, 233–243

    Google Scholar 

  24. Küronya A., Asymptotic Cohomological Functions on Projective Varieties, PhD thesis, University of Michigan, 2004

  25. Küronya A., Asymptotic cohomological functions on projective varieties, Amer. J. Math., 2006, 128(6), 1475–1519

    Article  MathSciNet  MATH  Google Scholar 

  26. Küronya A., Positivity on subvarieties and vanishing of higher cohomology, preprint available at http://arxiv.org/abs/1012.1102

  27. Lazarsfeld R., Positivity in Algebraic Geometry. I, Ergeb. Math. Grenzgeb. (3), 48, Springer, Berlin, 2004

    Google Scholar 

  28. Lazarsfeld R., Positivity in Algebraic Geometry. II, Ergeb. Math. Grenzgeb. (3), 49, Springer, Berlin, 2004

    Google Scholar 

  29. Mumford D., Abelian Varieties, Tata Inst. Fund. Res. Studies in Math., 5, Hindustan Book Agency, New Delhi, 2008

    MATH  Google Scholar 

  30. Nakayama N., Zariski-Decomposition and Abundance, MSJ Mem., 14, Mathematical Society of Japan, Tokyo, 2004

    MATH  Google Scholar 

  31. Siu Y.T., Some recent results in complex manifold theory related to vanishing theorems for the semipositive case, In: Arbeitstagung Bonn 1984, Max-Planck-Institut für Mathematik, Bonn, June 15–22, 1984, Lecture Notes in Math., 1111, Springer, Berlin, 1985, 169–192

    Google Scholar 

  32. Takayama S., Pluricanonical systems on algebraic varieties of general type, Invent. Math., 2006, 165(3), 551–587

    Article  MathSciNet  MATH  Google Scholar 

  33. Tanimoto S., personal communication, 2009

  34. Totaro B., Line bundles with partially vanishing cohomology, preprint available at http://arxiv.org/abs/1007.3955

  35. Tsuji H., Pluricanonical systems of projective varieties of general type. I, Osaka J. Math., 2006, 43(4), 967–995

    MathSciNet  MATH  Google Scholar 

  36. Tsuji H., Pluricanonical systems of projective varieties of general type. II, Osaka J. Math., 2007, 44(3), 723–764

    MathSciNet  MATH  Google Scholar 

  37. Zariski O., The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math., 1962, 76(3), 560–615

    Article  MathSciNet  MATH  Google Scholar 

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  1. Fordham University, 441 East Fordham Road, Bronx, NY, 10458, USA

    Michael A. Burr

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  1. Michael A. Burr
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Correspondence to Michael A. Burr.

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Burr, M.A. Asymptotic purity for very general hypersurfaces of ℙn × ℙn of bidegree (k, k). centr.eur.j.math. 10, 530–542 (2012). https://doi.org/10.2478/s11533-011-0126-8

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  • DOI: https://doi.org/10.2478/s11533-011-0126-8

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