Algebra & Number Theory

A generalized Bogomolov–Gieseker inequality for the three-dimensional projective space

Emanuele Macrì

Full-text: Open access

Abstract

A generalized Bogomolov–Gieseker inequality for tilt-stable complexes on a smooth projective threefold was conjectured by Bayer, Toda, and the author. We show that such inequality holds true in general if it holds true when the polarization is sufficiently small. As an application, we prove it for the three-dimensional projective space.

Article information

Source
Algebra Number Theory, Volume 8, Number 1 (2014), 173-190.

Dates
Received: 30 July 2012
Revised: 10 June 2013
Accepted: 11 June 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730138

Digital Object Identifier
doi:10.2140/ant.2014.8.173

Mathematical Reviews number (MathSciNet)
MR3207582

Zentralblatt MATH identifier
1308.14016

Subjects
Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 18E30: Derived categories, triangulated categories 14J30: $3$-folds [See also 32Q25]

Keywords
Bridgeland stability conditions derived category Bogomolov–Gieseker inequality

Citation

Macrì, Emanuele. A generalized Bogomolov–Gieseker inequality for the three-dimensional projective space. Algebra Number Theory 8 (2014), no. 1, 173--190. doi:10.2140/ant.2014.8.173. https://projecteuclid.org/euclid.ant/1513730138


Export citation

References

  • D. Arcara and A. Bertram, “Bridgeland-stable moduli spaces for $K$-trivial surfaces”, J. Eur. Math. Soc. 15:1 (2013), 1–38.
  • D. Arcara, A. Bertram, I. Coskun, and J. Huizenga, “The minimal model program for the Hilbert scheme of points on $\mathbb{P}\sp 2$ and Bridgeland stability”, Adv. Math. 235 (2013), 580–626.
  • A. Bayer, “A tour to stability conditions on derived categories”, notes, 2011, http://www.maths.ed.ac.uk/~abayer/dc-lecture-notes.pdf.
  • A. Bayer and E. Macr\`\i, “Projectivity and birational geometry of Bridgeland moduli spaces”, preprint, 2012. To appear in J. Am. Math. Soc.
  • A. Bayer, A. Bertram, E. Macr\`\i, and Y. Toda, “Bridgeland stability conditions on threefolds, II: An application to Fujita's conjecture”, preprint, 2011. To appear in J. Alg. Geom.
  • A. Bayer, E. Macr\`\i, and Y. Toda, “Bridgeland stability conditions on threefolds, I: Bogomolov–Gieseker type inequalities”, preprint, 2011. To appear in J. Alg. Geom.
  • T. Bridgeland, “Stability conditions on triangulated categories”, Ann. of Math. $(2)$ 166:2 (2007), 317–345.
  • T. Bridgeland, “Stability conditions on $K3$ surfaces”, Duke Math. J. 141:2 (2008), 241–291.
  • T. Bridgeland, “Spaces of stability conditions”, pp. 1–21 in Algebraic geometry, Part 1 (Seattle, 2005), edited by D. Abramovich et al., Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, RI, 2009.
  • D. Happel, I. Reiten, and S. O. Smalø, “Tilting in abelian categories and quasitilted algebras”, Mem. Amer. Math. Soc. 120:575 (1996), 1–88.
  • J. Harris, “The genus of space curves”, Math. Ann. 249:3 (1980), 191–204. http://msp.org/idx/mr/81i:14022MR 81i:14022
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
  • R. Hartshorne, “Stable vector bundles of rank $2$ on $\mathbb{P}\sp{3}$”, Math. Ann. 238:3 (1978), 229–280.
  • D. Huybrechts, “Introduction to stability conditions”, preprint, 2012. http://msp.org/idx/arx/1111.1745v2arXiv 1111.1745v2
  • J. Lo and Z. Qin, “Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces”, preprint, 2011.
  • A. Maciocia, “Computing the walls associated to Bridgeland stability conditions on projective surfaces”, preprint, 2012. To appear in Asian J. Math.
  • A. Maciocia and C. Meachan, “Rank-$1$ Bridgeland stable moduli spaces on a principally polarized abelian surface”, Int. Math. Res. Not. 2013:9 (2013), 2054–2077.
  • E. Macr\`\i, “Stability conditions on curves”, Math. Res. Lett. 14:4 (2007), 657–672.
  • H. Minamide, S. Yanagida, and K. Yoshioka, “Some moduli spaces of Bridgeland's stability conditions”, preprint, 2011.
  • A. Polishchuk, “Phases of Lagrangian-invariant objects in the derived category of an abelian variety”, preprint, 2012.
  • Y. Toda, “Introduction and open problems of Donaldson–Thomas theory”, pp. 289–318 in Derived categories in algebraic geometry (Tokyo, 2011), edited by Y. Kawamata, Eur. Math. Soc., Zürich, 2012.
  • Y. Toda, “Stability conditions and birational geometry of projective surfaces”, preprint, 2012. To appear in Compos. Math.
  • Y. Toda, “Bogomolov–Gieseker-type inequality and counting invariants”, J. Topol. 6:1 (2013), 217–250.
  • Y. Toda, “Stability conditions and extremal contractions”, Math. Ann. 357:2 (2013), 631–685.
  • S. Yanagida and K. Yoshioka, “Bridgeland's stabilities on abelian surfaces”, preprint, 2012.
  • K. Yoshioka, “Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface”, preprint, 2012.