Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T00:56:23.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability conditions and birational geometry of projective surfaces

Part of: Abelian categories Birational geometry

Published online by Cambridge University Press:  17 July 2014

Yukinobu Toda
Yukinobu Toda*
Affiliation:
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 email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

Keywords

Bridgeland stability conditionsminimal model program

MSC classification

Secondary: 18E30: Derived categories, triangulated categories 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 10 , October 2014 , pp. 1755 - 1788
Copyright
© The Author 2014 

References

Arcara, D. and Bertram, A., Bridgeland-stable moduli spaces for K-trivial surfaces. With an appendix by Max Lieblich, J. Eur. Math. Soc. 15 (2013), 1–38.Google Scholar
Arcara, D., Bertram, A., Coskun, I. and Huizenga, J., The minimal model program for Hilbert schemes of points on the projective plane and Bridgeland stability, Adv. Math. 235 (2013), 580–626.CrossRefGoogle Scholar
Bayer, A. and Macri, E., Projectivity and birational geometry of Bridgeland moduli spaces, J. Amer. Math. Soc. 27 (2014), 707–752.CrossRefGoogle Scholar
Bayer, A., Macri, E. and Toda, Y., Bridgeland stability conditions on 3-folds I: Bogomolov–Gieseker type inequalities, J. Algebraic Geom. 23 (2014), 117–163.Google Scholar
Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers. Analysis and topology on singular spaces I, Asterisque 100 (1982), 5–171.Google Scholar
Bridgeland, T., Flops and derived categories, Invent. Math. 147 (2002), 613–632.CrossRefGoogle Scholar
Bridgeland, T., Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), 317–345.Google Scholar
Bridgeland, T., Stability conditions on K3 surfaces, Duke Math. J. 141 (2008), 241–291.CrossRefGoogle Scholar
Van den Bergh, M., Three dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), 423–455.Google Scholar
Douglas, M., Dirichlet branes, homological mirror symmetry, and stability, in Proceedings of the ICM (Beijing 2002), Vol. III (Higher Education Press, Beijing, 2002), 395–408.Google Scholar
Gelfand, S. and Manin, Y., Methods of homological algebra, Springer Monographs in Mathematics, second edition (Springer, Berlin, 2003).CrossRefGoogle Scholar
Happel, D., Reiten, I. and Smalø, S. O., Tilting in abelian categories and quasitilted algebras, Memoirs of the American Mathematical Society, vol. 120 (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar
Huybrechts, D., Macri, E. and Stellari, P., Stability conditions for generic K3 categories, Compositio Math. 144 (2008), 134–162.CrossRefGoogle Scholar
Inaba, M., Toward a definition of moduli of complexes of coherent sheaves on a projective scheme, J. Math. Kyoto Univ. 42 (2002), 317–329.Google Scholar
Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, Preprint (2008), arXiv:0811.2435.Google Scholar
Maciocia, A., Computing the walls associated to Bridgeland stability conditions on projective surfaces, Preprint (2012), arXiv:1202.4587.Google Scholar
Maciocia, A. and Meachan, C., Rank one Bridgeland stable moduli spaces on a principally polarized abelian surfaces, Int. Math. Res. Not. IMRN 9 (2013), 2054–2077.CrossRefGoogle Scholar
Minamide, H., Yanagida, S. and Yoshioka, K., Fourier–Mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions, Preprint (2011), arXiv:1106.5217.Google Scholar
Minamide, H., Yanagida, S. and Yoshioka, K., Some moduli spaces of Bridgeland stability conditions, Preprint (2011), arXiv:1111.6187.Google Scholar
Toda, Y., Curve counting theories via stable objects II. DT/ncDT/flop formula, J. Reine Angew. Math. 675 (2013), 1–51.Google Scholar
Toda, Y., Stability conditions and extremal contractions, Math. Ann. 357 (2013), 631–685.Google Scholar
Toda, Y., Moduli stacks and invariants of semistable objects on K3 surfaces, Adv. Math. 217 (2008), 2736–2781.CrossRefGoogle Scholar
Toda, Y., Stability conditions and crepant small resolutions, Trans. Amer. Math. Soc. 360 (2008), 6149–6178.Google Scholar
Toda, Y., Curve counting theories via stable objects I: DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), 1119–1157.Google Scholar
Yanagida, S. and Yoshioka, K., Bridgeland stabilities on abelian surfaces, Preprint (2012),arXiv:1203.0884.Google Scholar