Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Moduli of PT-semistable objects II

Author: Jason Lo
Journal: Trans. Amer. Math. Soc. 365 (2013), 4539-4573
MSC (2010): Primary 14F05, 14D20, 18E30, 14J60; Secondary 14J30
Published electronically: March 5, 2013
MathSciNet review: 3066765
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We generalise the techniques of semistable reduction for flat families of sheaves to the setting of the derived category $ D^b(X)$ of coherent sheaves on a smooth projective three-fold $ X$. Then we construct the moduli of PT-semistable objects in $ D^b(X)$ as an Artin stack of finite type that is universally closed. In the absence of strictly semistable objects, we construct the moduli as a proper algebraic space of finite type.

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

  • [AP] 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),
  • [ABL] D. Arcara, A. Bertram and M. Lieblich, Bridgeland-stable moduli spaces for K-trivial surfaces, arXiv:0708.2247v1 [math.AG].
  • [Bay] Arend Bayer, Polynomial Bridgeland stability conditions and the large volume limit, Geom. Topol. 13 (2009), no. 4, 2389-2425. MR 2515708 (2010h:14026),
  • [BV] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1-36, 258 (English, with English and Russian summaries). MR 1996800 (2004h:18009)
  • [GM] Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR 1438306 (97j:18001)
  • [HRS] 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)
  • [Huy] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2006. MR 2244106 (2007f:14013)
  • [HL] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870 (98g:14012)
  • [Ina] Michi-aki Inaba, Toward a definition of moduli of complexes of coherent sheaves on a projective scheme, J. Math. Kyoto Univ. 42 (2002), no. 2, 317-329. MR 1966840 (2004e:14022)
  • [KS1] Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076 (2006k:18001)
  • [KS2] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006 (92a:58132)
  • [Knu] Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin, 1971. MR 0302647 (46 #1791)
  • [Lan] Stacy G. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. (2) 101 (1975), 88-110. MR 0364255 (51 #510)
  • [Lo] Jason Lo, Moduli of PT-semistable objects I, J. Algebra 339 (2011), 203-222. MR 2811320 (2012h:14037),
  • [LMB] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 39, Springer-Verlag, Berlin, 2000.
  • [Lie] Max Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), no. 1, 175-206. MR 2177199 (2006f:14009),
  • [Mar] Masaki Maruyama, On boundedness of families of torsion free sheaves, J. Math. Kyoto Univ. 21 (1981), no. 4, 673-701. MR 637512 (83a:14019)
  • [Orl] D. O. Orlov, Equivalences of derived categories and $ K3$ surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361-1381. Algebraic geometry, 7. MR 1465519 (99a:14054),
  • [PT] R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407-447. MR 2545686 (2010h:14089),
  • [Pol] 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)
  • [Pot] J. Le Potier, Faisceaux semi-stables et systèmes cohérents, Vector bundles in algebraic geometry (Durham, 1993) London Math. Soc. Lecture Note Ser., vol. 208, Cambridge Univ. Press, Cambridge, 1995, pp. 179-239 (French, with French summary). MR 1338417 (96h:14010)
  • [Tod1] Yukinobu Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157-208. MR 2541209 (2011b:14043),
  • [Tod2] Yukinobu Toda, Moduli stacks and invariants of semistable objects on $ K3$ surfaces, Adv. Math. 217 (2008), no. 6, 2736-2781. MR 2397465 (2009a:14017),

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14F05, 14D20, 18E30, 14J60, 14J30

Retrieve articles in all journals with MSC (2010): 14F05, 14D20, 18E30, 14J60, 14J30

Additional Information

Jason Lo
Affiliation: Department of Mathematics, Building 380, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematics, 202 Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65211

Keywords: PT-stability, semistable reduction, derived category, moduli, valuative criterion
Received by editor(s): November 30, 2010
Received by editor(s) in revised form: May 3, 2011
Published electronically: March 5, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society