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

Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu

doi:10.1090/S1056-3911-2013-00617-7

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
Full-text PDF

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

Daniele Arcara and Aaron Bertram. Reider’s theorem and Thaddeus pairs revisited, 2009. arXiv:0904.3500.

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, DOI https://doi.org/10.4171/JEMS/354

Paul S. Aspinwall and Michael R. Douglas, D-brane stability and monodromy, J. High Energy Phys. 5 (2002), no. 31, 35. MR 1915365, DOI https://doi.org/10.1088/1126-6708/2002/05/031

Paul S. Aspinwall and Albion Lawrence, Derived categories and zero-brane stability, J. High Energy Phys. 8 (2001), Paper 4, 27. MR 1867069, DOI https://doi.org/10.1088/1126-6708/2001/08/004

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, DOI https://doi.org/10.1515/CRELLE.2006.005

Paul S. Aspinwall, D-branes on Calabi-Yau manifolds, Progress in string theory, World Sci. Publ., Hackensack, NJ, 2005, pp. 1–152. MR 2405589

Arend Bayer, Polynomial Bridgeland stability conditions and the large volume limit, Geom. Topol. 13 (2009), no. 4, 2389–2425. MR 2515708, DOI https://doi.org/10.2140/gt.2009.13.2389

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.

Tom Bridgeland and Antony Maciocia, Fourier-Mukai transforms for $K3$ and elliptic fibrations, J. Algebraic Geom. 11 (2002), no. 4, 629–657. MR 1910263, DOI https://doi.org/10.1090/S1056-3911-02-00317-X

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, DOI https://doi.org/10.1215/00127094-1444249

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

Tom Bridgeland, Stability conditions on a non-compact Calabi-Yau threefold, Comm. Math. Phys. 266 (2006), no. 3, 715–733. MR 2238896, DOI https://doi.org/10.1007/s00220-006-0048-7

Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI https://doi.org/10.4007/annals.2007.166.317

Tom Bridgeland, Stability conditions on $K3$ surfaces, Duke Math. J. 141 (2008), no. 2, 241–291. MR 2376815, DOI https://doi.org/10.1215/S0012-7094-08-14122-5

Michael R. Douglas, Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 395–408. MR 1957548

Michael R. Douglas, Rene Reinbacher, and Shing-Tung Yau. Branes, bundles and attractors: Bogomolov and beyond, 2006. arXiv:math/0604597.

D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), no. 1, 77–85. MR 527826, DOI https://doi.org/10.2307/2373939

Joe Harris, The genus of space curves, Math. Ann. 249 (1980), no. 3, 191–204. MR 579101, DOI https://doi.org/10.1007/BF01363895

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

Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168

Daniel Huybrechts, Emanuele Macrì, and Paolo Stellari, Stability conditions for generic $K3$ categories, Compos. Math. 144 (2008), no. 1, 134–162. MR 2388559, DOI https://doi.org/10.1112/S0010437X07003065

Dieter Happel, Idun Reiten, and Sverre O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88. MR 1327209, DOI https://doi.org/10.1090/memo/0575

Akira Ishii, Kazushi Ueda, and Hokuto Uehara, Stability conditions on $A_n$-singularities, J. Differential Geom. 84 (2010), no. 1, 87–126. MR 2629510

M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508. MR 939472, DOI https://doi.org/10.1007/BF01393744

Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918

Maxim Kontsevich and Yan Soibelman. Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, 2008. arXiv:0811.2435.

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, DOI https://doi.org/10.1090/pspum/080.1/2483939

Indranil Biswas, On the fundamental group-scheme, Bull. Sci. Math. 133 (2009), no. 5, 477–483. MR 2538008, DOI https://doi.org/10.1016/j.bulsci.2008.11.002

Emanuele Macrì. Some examples of moduli spaces of stability conditions on derived categories, 2004. arXiv:math/0411613.

Emanuele Macrì, Stability conditions on curves, Math. Res. Lett. 14 (2007), no. 4, 657–672. MR 2335991, DOI https://doi.org/10.4310/MRL.2007.v14.n4.a10

S. Meinhardt, Stability conditions on generic complex tori, Internat. J. Math. 23 (2012), no. 5, 1250035, 17. MR 2914652, DOI https://doi.org/10.1142/S0129167X12500358

So Okada, Stability manifold of ${\Bbb P}^1$, J. Algebraic Geom. 15 (2006), no. 3, 487–505. MR 2219846, DOI https://doi.org/10.1090/S1056-3911-06-00432-2

Christian Okonek and Heinz Spindler, Das Spektrum torsionsfreier Garben. II, Seminar on deformations (Łódź/Warsaw, 1982/84) Lecture Notes in Math., vol. 1165, Springer, Berlin, 1985, pp. 211–234 (German, with English summary). MR 825758, DOI https://doi.org/10.1007/BFb0076156

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, DOI https://doi.org/10.17323/1609-4514-2007-7-1-109-134

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

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, DOI https://doi.org/10.1515/crll.1980.315.211

Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076

Yukinobu Toda, Stability conditions and crepant small resolutions, Trans. Amer. Math. Soc. 360 (2008), no. 11, 6149–6178. MR 2425708, DOI https://doi.org/10.1090/S0002-9947-08-04509-1

Yukinobu Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157–208. MR 2541209, DOI https://doi.org/10.1215/00127094-2009-038

Yukinobu Toda, Stability conditions and Calabi-Yau fibrations, J. Algebraic Geom. 18 (2009), no. 1, 101–133. MR 2448280, DOI https://doi.org/10.1090/S1056-3911-08-00511-0

Yukinobu Toda, Curve counting theories via stable objects I. DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), no. 4, 1119–1157. MR 2669709, DOI 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
MR Author ID: 728427
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

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.