Abstract
In Bayer and Macrì (J Am Math Soc 27(3):707–752, 2014), the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X. In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on Y with compact support. We also show that the nef divisor is compatible with the polarising ample line bundle coming from the GIT construction of the moduli space in the special case when Y admits a tilting bundle and the stability condition arises from a \(\theta \)-stability condition for the endomorphism algebra. Our main tool generalises the work of Abramovich–Polishchuk (J Reine Angew Math 590:89–130, 2006) and Polishchuk (Mosc Math J 7(1):109–134, 2007): given a t-structure on the derived category \(D_c(Y)\) on Y of objects with compact support and a base scheme S, we construct a constant family of t-structures on a category of objects on \(Y \times S\) with compact support relative to S.
References
Arinkin, D., Bezrukavnikov, R.: Perverse coherent sheaves. Mosc. Math. J. 10(1), 3–29, 271 (2010)
Arcara, D., Bertram, A.: Bridgeland-stable moduli spaces for \(K\)-trivial surfaces. J. Eur. Math. Soc. (JEMS) 15(1), 1–38 (2013). (With an appendix by Max Lieblich)
Aspinwall, P. S., Bridgeland, T., Craw, A., Douglas, M.R., Gross, M., Kapustin, A., Moore, G.W., Segal, G., Szendrői, B., Wilson, P.M.H.: Dirichlet branes and mirror symmetry, volume 4 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2009)
Arcara, D., Bertram, A., Coskun, I., Huizenga, J.: The minimal model program for the Hilbert scheme of points on \(\mathbb{P}^2\) and Bridgeland stability. Adv. Math. 235, 580–626 (2013)
Anno, R., Bezrukavnikov, R., Mirković, I.: Stability conditions for Slodowy slices and real variations of stability (2011). arXiv:1108.1563
Abramovich, D., Polishchuk, A.: Sheaves of \(t\)-structures and valuative criteria for stable complexes. J. Reine Angew. Math. 590, 89–130 (2006)
Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras. Vol. 1, volume 65 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge. Techniques of representation theory (2006)
Bertram, A., Coskun, I.: The birational geometry of the Hilbert scheme of points on surfaces. Birational geometry, rational curves, and arithmetic, pp. 15–55. Springer, New York (2013)
Bolognese, B., Huizenga, J., Lin, Y., Riedl, E., Schmidt, B., Woolf, M., Zhao, X.: Nef cones of Hilbert schemes of points on surfaces (2015). arXiv:1509.04722
Butler, M., King, A.: Minimal resolutions of algebras. J. Algebra 212(1), 323–362 (1999)
Bayer, A., Macrì, E.: The space of stability conditions on the local projective plane. Duke Math. J. 160(2), 263–322 (2011)
Bayer, A., Macrì, E.: MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones. Lagrangian Fibrations. Invent. Math. 198(3), 505–590 (2014)
Bayer, A., Macrì, E.: Projectivity and birational geometry of Bridgeland moduli spaces. J. Am. Math. Soc. 27(3), 707–752 (2014)
Bayer, A., Macrì, E., Stellari, P.: The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014). arXiv:1410.1585
Bertram, A., Martinez, C., Wang, J.: The birational geometry of moduli space of sheaves on the projective plane (2013). arXiv:1301.2011
Bridgeland, T.: Stability conditions on a non-compact Calabi-Yau threefold. Commun. Math. Phys. 266(3), 715–733 (2006)
Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007)
Bridgeland, T.: Stability conditions on \(K3\) surfaces. Duke Math. J. 141(2), 241–291 (2008)
Bridgeland, T.: Stability conditions and Kleinian singularities. Int. Math. Res. Not. (IMRN) 21, 4142–4157 (2009)
Bridgeland, T., Stern, D.: Helices on del Pezzo surfaces and tilting Calabi-Yau algebras. Adv. Math. 224(4), 1672–1716 (2010)
Bocklandt, R., Schedler, T., Wemyss, M.: Superpotentials and higher order derivations. J. Pure Appl. Algebra 214(9), 1501–1522 (2010)
Brav, C., Thomas, H.: Braid groups and Kleinian singularities. Math. Ann. 351(4), 1005–1017 (2011)
Choi, J., Chung, K.: The geometry of the moduli space of one-dimensional sheaves. Sci. China Math. 58(3), 487–500 (2015)
Coskun, I., Huizenga, J.: The ample cone of moduli spaces of sheaves on the plane (2014). arXiv:1409.5478
Coskun, I., Huizenga, J.: Interpolation, Bridgeland stability and monomial schemes in the plane. J. Math. Pures Appl. (9) 102(5), 930–971 (2014)
Coskun, I., Huizenga, J.: The birational geometry of the moduli spaces of sheaves on \(\mathbb{P}^2\). In: Proceedings of the Gökova Geometry-Topology Conference 2014, pp. 114–155. Gökova Geometry/Topology Conference (GGT), Gökova (2015)
Coskun, I., Huizenga, J.: The nef cone of the moduli space of sheaves and strong Bogomolov inequalities (2015). arXiv:1512.02661
Coskun, I., Huizenga, J., Woolf, M.: The effective cone of the moduli space of sheaves on the plane (2014). arXiv:1401.1613
Craw, A., Ishii, A.: Flops of \(G\)-Hilb and equivalences of derived categories by variation of GIT quotient. Duke Math. J. 124(2), 259–307 (2004)
Eisenbud, D.: Commutative Algebra, Volume 150 of Graduate Texts in Mathematics. Springer, New York (1995). (With a view toward algebraic geometry)
Hassett, B., Tschinkel, Y.: Extremal rays and automorphisms of holomorphic symplectic varieties (2015). arXiv:1506.08153
Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006)
Hille, L., Van den Bergh, M.: Fourier-Mukai transforms. In: Handbook of tilting theory, volume 332 of London Math. Soc. Lecture Note Ser., pp. 147–177. Cambridge Univ. Press, Cambridge (2007)
Ishii, A., Ueda, K.: Dimer models and crepant resolutions. To appear in Hokkaido Mathematical Journal (2013). arXiv:1303.4028
Ishii, A., Ueda, K., Uehara, H.: Stability conditions on \(A_n\)-singularities. J. Differ. Geom. 84(1), 87–126 (2010)
Karmazyn, J.: Quiver GIT for varieties with tilting bundles (2014). arXiv:1407.5005
King, A.: Moduli of representations of finite-dimensional algebras. Q. J. Math. Oxford Ser. (2) 45(180), 515–530 (1994)
King, A.: Tilting bundles on some rational surfaces (1997). Unpublished http://www.maths.bath.ac.uk/~masadk/papers/tilt
Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. 39(1), 19–55 (1976)
Lieblich, M.: Moduli of complexes on a proper morphism. J. Algebraic Geom. 15(1), 175–206 (2006)
Li, C., Zhao, X.: The MMP for deformations of Hilbert schemes of points on the projective plane (2013). arXiv:1312.1748
Li, C., Zhao, X.: Birational models of moduli spaces of coherent sheaves on the projective plane (2016). arXiv:1603.05035
Manivel, L.: Théorèmes d’annulation sur certaines variétés projectives. Comment. Math. Helv. 71(3), 402–425 (1996)
Miličić, D.: Lectures on Derived Categories (2003). Unpublished lecture notes http://www.math.utah.edu/~milicic/Eprints/dercat
Minamide, H., Yanagida, S., Yoshioka, K.: Fourier-mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions (2011). arXiv:1106.5217
Minamide, H., Yanagida, S., Yoshioka, K.: Some moduli spaces of Bridgeland’s stability conditions. Int. Math. Res. Not. 19, 5264–5327 (2014)
Meachan, C., Zhang, Z.: Birational geometry of singular moduli spaces of O’Grady type. Adv. Math. 296, 210–267 (2016)
Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9(1), 205–236 (1996)
Neeman, A.: Triangulated Categories, Volume 148 of Annals of Mathematics Studies. Princeton University Press, Princeton (2001)
Nuer, H.: Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface (2014). arXiv:1406.0908
Orlov, D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Math. 56(4), 852–862 (1992)
Polishchuk, A.: Constant families of \(t\)-structures on derived categories of coherent sheaves. Mosc. Math. J. 7(1), 109–134 (2007). 167
Rouquier, R.: Derived categories and algebraic geometry. In: Triangulated Categories, Volume 375 of London Math. Soc. Lecture Note Ser., pp 351–370. Cambridge Univ. Press, Cambridge (2010)
Théorie des intersections et théorème de Riemann-Roch. (French) Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. Lecture Notes in Mathematics, vol. 225, xii\(+\)700, pp. 14–06. Springer, Berlin, New York (1971)
The Stacks Project Authors: Stacks project (2016). http://stacks.math.columbia.edu. Accessed 29 Sept 2016
Thomas, R. P.: Stability conditions and the braid group. In: Superstring Theory, Volume 1 of Adv. Lect. Math. (ALM), pp 209–233. Int. Press, Somerville, MA (2008)
Toda, Y.: Stability conditions and crepant small resolutions. Trans. Am. Math. Soc. 360(11), 6149–6178 (2008)
Toda, Y.: Stability conditions and Calabi-Yau fibrations. J. Algebraic Geom. 18(1), 101–133 (2009)
van den Bergh, M.: Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004)
Matthew W.: Nef and effective cones on the moduli space of torsion sheaves on the projective plane (2013). arXiv:1305.1465
Yoshioka, K.: Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface (2012). arXiv:1206.4838
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of Johan Louis Dupont
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bayer, A., Craw, A. & Zhang, Z. Nef divisors for moduli spaces of complexes with compact support. Sel. Math. New Ser. 23, 1507–1561 (2017). https://doi.org/10.1007/s00029-016-0298-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-016-0298-y
Keywords
- Bridgeland stability conditions
- Derived categories
- t-structures
- Moduli spaces of sheaves and complexes
- Nef divisors
Mathematics Subject Classification
- Primary 14D20
- Secondary 14F05
- 14J28
- 18E30