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Stability conditions and extremal contractions

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Abstract

We study extremal contractions from smooth projective varieties via a moduli theoretic approach. In the two dimensional case, we show that any extremal contraction appears as a moduli space of Bridgeland stable objects in the derived category of coherent sheaves. In the three dimensional case, we show that a a similar result holds with respect to conjectural Bridgeland stability conditions.

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Notes

  1. The notion of S-equivalence is a direct analogue used in the study of moduli of sheaves. See Definition 2.2.

  2. This means that \(\mathcal{P }_{B, f^{*}\omega }(1)\) is a noetherian and artinian abelian category.

  3. A stability condition in Definition 2.1 was called numerical stability condition in [11]. We omit ‘numerical’ since we do not deal with non-numerical stability conditions.

  4. The locally finiteness of \(\sigma _0\) is obvious since the image of \(Z_{f^{*}\omega }\) is a discrete subgroup.

  5. This was denoted by \(\mathop {^{{p}}{\mathrm{Per}}}\nolimits (X/\fancyscript{A}_0)\) in [33].

  6. Here we use the different perversity from the surface case.

  7. Kawamata informed the author that he later proved the minimality of \(\mathcal{D }_Y\).

References

  1. Arcara, D., Bertram, A.: Bridgeland-stable moduli spaces for K-trivial surfaces (preprint). arXiv:0708.2247

  2. Arcara, D., Bertram, A., Coskun, I., Huizenga, J.: The minimal model program for Hilbert schemes of points on the projective plane and Bridgeland stability (preprint). arXiv:1203.0316

  3. Bayer, A.: Polynomial Bridgeland stability conditions and the large volume limit. Geom. Topol. 13, 2389–2425 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bayer, A., Macri, E.: Projectivity and birational geometry of Bridgeland moduli spaces (preprint). arXiv:1203.4613

  5. Bayer, A., Macri, E.: The space of stability conditions on the local projective plane. Duke. Math. J. 160, 263–322 (2011)

    Google Scholar 

  6. Bayer, A., Macri, E., Toda, Y.: Bridgeland stability conditions on 3-folds I: Bogomolov-Gieseker type inequalities. JAG (to appear)

  7. Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23, 405–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bogomolov, F.A.: Holomorphic tensors and vector bundles on projective manifolds. Izv. Akad. Nauk SSSR Ser. Math. 42, 1227–1287 (1978)

    MathSciNet  Google Scholar 

  9. Bondal, A., Orlov, D.: Semiorthgonal decomposition for algebraic varieties (preprint). arXiv:9506012

  10. Bridgeland, T.: Flops and derived categories. Invent. Math. 147, 613–632 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. 166, 317–345 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bridgeland, T.: Stability conditions on \({K}\)3 surfaces. Duke Math. J. 141, 241–291 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Van den Bergh, M.: Three dimensional flops and noncommutative rings. Duke Math. J. 122, 423–455 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Douglas, M.: Dirichlet branes, homological mirror symmetry, and stability. In: Proceedings of the 1998 ICM, pp. 395–408 (2002)

  15. Gieseker, D.: On a theorem of Bogomolov on Chern classes of stable bundles. Am. J. Math. 101, 77–85 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras, vol. 120 of Mem. Amer. Math. Soc. (1996)

  17. Huybrechts, D., Lehn, M.: Geometry of Moduli Spaces of Sheaves, Vol. E31 of Aspects in Mathematics. Vieweg, Braunschweig (1997)

  18. Inaba, M.: Toward a definition of moduli of complexes of coherent sheaves on a projective scheme. J. Math. Kyoto Univ. 42(2), 317–329 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Kawamata, Y.: Log crepant birational maps and derived categories. J. Math. Sci. Univ. Tokyo 12, 1–53 (2005)

    MathSciNet  Google Scholar 

  20. Kawamata, Y.: Derived categories and birational geometry. Proc. Sympos. Pure Math. 80, 655–665 (2009)

    Article  MathSciNet  Google Scholar 

  21. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Vol. 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998)

  22. Langer, A.: Semistable sheaves in positive characteristic. Ann. Math. 159, 251–276 (2004)

    Article  MATH  Google Scholar 

  23. Maciocia, A.: Computing the walls associated to Bridgeland stability conditions on projective surfaces (preprint). arXiv:1202.4587

  24. Maciocia, A., Meachan, C.: Rank one Bridgeland stable moduli spaces on a principally polarized abelian surfaces (preprint). arXiv:1107.5304

  25. Matsuki, K., Wentworth, R.: Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface. Int. J. Math. 8, 97–148 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Minamide, H., Yanagida, S., Yoshioka, K.: Fourier-Mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions (preprint). arXiv:1106.5217

  27. Minamide, H., Yanagida, S., Yoshioka, K.: Some moduli spaces of Bridgeland stability conditions (preprint). arXiv:1111.6187

  28. Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116, 133–176 (1982)

    Article  MATH  Google Scholar 

  29. Thaddeus, M.: Geometric invariant theory and flips. J. Am. Math. Soc. 9, 691–723 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Toda, Y.: Moduli stacks and invariants of semistable objects on K3 surfaces. Adv. Math. 217, 2736–2781 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Toda, Y.: Bogomolov-Gieseker type inequality and counting invariants (preprint). arXiv:1112.3411

  32. Toda, Y.: Curve counting theories via stable objects I: DT/PT correspondence. J. Am. Math. Soc. 23, 1119–1157 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Toda, Y., Uehara, H.: Tilting generators via ample line bundles. Adv. Math. 223, 1–29 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yanagida, S., Yoshioka, K.: Bridgeland stabilities on abelian surfaces (preprint). arXiv:1203.0884

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Acknowledgments

The author is grateful to Arend Bayer for valuable comments. The proof of Proposition 3.13 is benefited by the communication with Emanuele Macri. This work is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan. This work is also supported by Grant-in Aid for Scientific Research grant (22684002), and partly (S-19104002), from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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  1. Todai Institute for Advanced Studies (TODIAS), Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan

    Yukinobu Toda

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  1. Yukinobu Toda
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Correspondence to Yukinobu Toda.

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Dedicated to Professor Yujiro Kawamata on the occasion of his 60-th birthday.

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Toda, Y. Stability conditions and extremal contractions. Math. Ann. 357, 631–685 (2013). https://doi.org/10.1007/s00208-013-0915-4

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  • DOI: https://doi.org/10.1007/s00208-013-0915-4

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