Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities


Authors: Arend Bayer, Emanuele Macrì and Yukinobu Toda
Journal: J. Algebraic Geom. 23 (2014), 117-163
DOI: https://doi.org/10.1090/S1056-3911-2013-00617-7
Published electronically: July 29, 2013
MathSciNet review: 3121850
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Abstract | References | Additional Information

Abstract: We construct new t-structures on the derived category of coherent
sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a Bogomolov-Gieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples.

Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.


References [Enhancements On Off] (What's this?)

  • [AB09] Daniele Arcara and Aaron Bertram.
    Reider's theorem and Thaddeus pairs revisited, 2009.
    arXiv:0904.3500.
  • [ABL07] Daniele Arcara and Aaron Bertram, Bridgeland-stable moduli spaces for $ K$-trivial surfaces, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 1-38. With an appendix by Max Lieblich. MR 2998828, https://doi.org/10.4171/JEMS/354
  • [AD02] Paul S. Aspinwall and Michael R. Douglas, D-brane stability and monodromy, J. High Energy Phys. 5 (2002), no. 31, 35. MR 1915365 (2004d:81081), https://doi.org/10.1088/1126-6708/2002/05/031
  • [AL01] Paul S. Aspinwall and Albion Lawrence, Derived categories and zero-brane stability, J. High Energy Phys. 8 (2001), Paper 4, 27. MR 1867069 (2003a:81181), https://doi.org/10.1088/1126-6708/2001/08/004
  • [AP06] Dan Abramovich and Alexander Polishchuk, Sheaves of $ t$-structures and valuative criteria for stable complexes, J. Reine Angew. Math. 590 (2006), 89-130. MR 2208130 (2007g:14014), https://doi.org/10.1515/CRELLE.2006.005
  • [Asp05] Paul S. Aspinwall.
    D-branes on Calabi-Yau manifolds.
    In Progress in string theory, pages 1-152. World Sci. Publ., Hackensack, NJ, 2005.
    arXiv:hep-th/0403166. MR 2405589 (2009i:81145)
  • [Bay09] Arend Bayer, Polynomial Bridgeland stability conditions and the large volume limit, Geom. Topol. 13 (2009), no. 4, 2389-2425. MR 2515708 (2010h:14026), https://doi.org/10.2140/gt.2009.13.2389
  • [BBMT11] Arend Bayer, Aaron Bertram, Emanuele Macrì, and Yukinobu Toda.
    Bridgeland stability conditions on threefolds II: An application to Fujita's conjecture, 2011.
    arXiv:1106.3430.
  • [BM02] Tom Bridgeland and Antony Maciocia, Fourier-Mukai transforms for $ K3$ and elliptic fibrations, J. Algebraic Geom. 11 (2002), no. 4, 629-657. MR 1910263 (2004e:14019), https://doi.org/10.1090/S1056-3911-02-00317-X
  • [BM11] Arend Bayer and Emanuele Macrì, The space of stability conditions on the local projective plane, Duke Math. J. 160 (2011), no. 2, 263-322. MR 2852118 (2012k:14019), https://doi.org/10.1215/00127094-1444249
  • [Bog78] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1227-1287, 1439 (Russian). MR 522939 (80j:14014)
  • [Bri06] Tom Bridgeland, Stability conditions on a non-compact Calabi-Yau threefold, Comm. Math. Phys. 266 (2006), no. 3, 715-733. MR 2238896 (2007d:14075), https://doi.org/10.1007/s00220-006-0048-7
  • [Bri07] Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317-345. MR 2373143 (2009c:14026), https://doi.org/10.4007/annals.2007.166.317
  • [Bri08] Tom Bridgeland, Stability conditions on $ K3$ surfaces, Duke Math. J. 141 (2008), no. 2, 241-291. MR 2376815 (2009b:14030), https://doi.org/10.1215/S0012-7094-08-14122-5
  • [Dou02] Michael R. Douglas, Dirichlet branes, homological mirror symmetry, and stability, (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 395-408. MR 1957548 (2004c:81200)
  • [DRY06] Michael R. Douglas, Rene Reinbacher, and Shing-Tung Yau.
    Branes, bundles and attractors: Bogomolov and beyond, 2006.
    arXiv:math/0604597.
  • [Gie79] D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), no. 1, 77-85. MR 527826 (80j:14015), https://doi.org/10.2307/2373939
  • [Har80] Joe Harris, The genus of space curves, Math. Ann. 249 (1980), no. 3, 191-204. MR 579101 (81i:14022), https://doi.org/10.1007/BF01363895
  • [Har94] Robin Hartshorne, The genus of space curves, Ann. Univ. Ferrara Sez. VII (N.S.) 40 (1994), 207-223 (1996) (English, with English and Italian summaries). MR 1399631 (97d:14044)
  • [HL10] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168 (2011e:14017)
  • [HMS08] Daniel Huybrechts, Emanuele Macrì, and Paolo Stellari, Stability conditions for generic $ K3$ categories, Compos. Math. 144 (2008), no. 1, 134-162. MR 2388559 (2009a:14050), https://doi.org/10.1112/S0010437X07003065
  • [HRS96] Dieter Happel, Idun Reiten, and SmaløSverre O., Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88. MR 1327209 (97j:16009)
  • [IUU10] Akira Ishii, Kazushi Ueda, and Hokuto Uehara, Stability conditions on $ A_n$-singularities, J. Differential Geom. 84 (2010), no. 1, 87-126. MR 2629510 (2011f:14027)
  • [Kap88] M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479-508. MR 939472 (89g:18018), https://doi.org/10.1007/BF01393744
  • [Kon95] Maxim Kontsevich, Homological algebra of mirror symmetry, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120-139. MR 1403918 (97f:32040)
  • [KS08] Maxim Kontsevich and Yan Soibelman.
    Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, 2008.
    arXiv:0811.2435.
  • [Lan09a] Adrian Langer, Moduli spaces of sheaves and principal $ G$-bundles, Algebraic geometry--Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 273-308. MR 2483939 (2010d:14010)
  • [Lan09b] Adrian Langer.
    On the S-fundamental group scheme, 2009.
    arXiv:0905.4600. MR 2538008 (2010f:14049)
  • [Mac04] Emanuele Macrì.
    Some examples of moduli spaces of stability conditions on derived categories, 2004.
    arXiv:math/0411613.
  • [Mac07] Emanuele Macrì, Stability conditions on curves, Math. Res. Lett. 14 (2007), no. 4, 657-672. MR 2335991 (2008k:18011)
  • [Mei07] S. Meinhardt, Stability conditions on generic complex tori, Internat. J. Math. 23 (2012), no. 5, 1250035, 17. MR 2914652, https://doi.org/10.1142/S0129167X12500358
  • [Oka06] So Okada, Stability manifold of $ {\mathbb{P}}^1$, J. Algebraic Geom. 15 (2006), no. 3, 487-505. MR 2219846 (2007b:14036), https://doi.org/10.1090/S1056-3911-06-00432-2
  • [OS85] Christian Okonek and Heinz Spindler.
    Das Spektrum torsionsfreier Garben. II.
    In Seminar on deformations (Łódź/Warsaw, 1982/84), volume 1165 of Lecture Notes in Math., pages 211-234. Springer, Berlin, 1985. MR 825758 (87d:14012)
  • [Pol07] A. Polishchuk, Constant families of $ t$-structures on derived categories of coherent sheaves, Mosc. Math. J. 7 (2007), no. 1, 109-134, 167 (English, with English and Russian summaries). MR 2324559 (2008e:14020)
  • [Rei78] Miles Reid, Bogomolov's theorem $ c_{1}^{2}\leq 4c_{2}$, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 623-642. MR 578877 (82b:14014)
  • [Sch80] Michael Schneider, Chernklassen semi-stabiler Vektorraumbündel vom Rang $ 3$ auf dem komplex-projektiven Raum, J. Reine Angew. Math. 315 (1980), 211-220 (German). MR 564534 (81f:14008), https://doi.org/10.1515/crll.1980.315.211
  • [Sim92] Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95. MR 1179076 (94d:32027)
  • [Tod08] Yukinobu Toda, Stability conditions and crepant small resolutions, Trans. Amer. Math. Soc. 360 (2008), no. 11, 6149-6178. MR 2425708 (2010b:14079), https://doi.org/10.1090/S0002-9947-08-04509-1
  • [Tod09a] Yukinobu Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157-208. MR 2541209 (2011b:14043), https://doi.org/10.1215/00127094-2009-038
  • [Tod09b] Yukinobu Toda, Stability conditions and Calabi-Yau fibrations, J. Algebraic Geom. 18 (2009), no. 1, 101-133. MR 2448280 (2009k:14076), https://doi.org/10.1090/S1056-3911-08-00511-0
  • [Tod10] Yukinobu Toda, Curve counting theories via stable objects I. DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), no. 4, 1119-1157. MR 2669709 (2011i:14020), https://doi.org/10.1090/S0894-0347-10-00670-3


Additional Information

Arend Bayer
Affiliation: Department of Mathematics, University of Connecticut U-3009, 196 Auditorium Road, Storrs, Connecticut 06269-3009
Address at time of publication: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh Scotland EH9 3JZ United Kingdom
Email: bayer@math.uconn.edu

Emanuele Macrì
Affiliation: Mathematical Institute, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany & Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, Utah 84112
Address at time of publication: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210
Email: macri.6@math.osu.edu

Yukinobu Toda
Affiliation: Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
Email: yukinobu.toda@ipmu.jp

DOI: https://doi.org/10.1090/S1056-3911-2013-00617-7
Received by editor(s): January 23, 2011
Received by editor(s) in revised form: January 25, 2012
Published electronically: July 29, 2013
Additional Notes: The first author was partially supported by NSF grants DMS-0801356/DMS-1001056 and DMS-1101377. The second author was partially supported by NSF grant DMS-1001482/DMS-1160466, and by the Hausdorff Center for Mathematics, Bonn, and SFB/TR 45. The third author was supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and Grant-in Aid for Scientific Research grant (22684002), partly (S-19104002), from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
Article copyright: © Copyright 2013 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.

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