Bridgeland stability on threefolds: Some wall crossings
Author:
Benjamin Schmidt
Journal:
J. Algebraic Geom. 29 (2020), 247-283
DOI:
https://doi.org/10.1090/jag/752
Published electronically:
November 15, 2019
MathSciNet review:
4069650
Full-text PDF
Abstract |
References |
Additional Information
Abstract: Following up on the construction of Bridgeland stability condition on $\mathbb {P}^3$ by Macrì, we develop techniques to study concrete wall crossing behavior for the first time on a threefold. In some cases, such as complete intersections of two hypersurfaces of the same degree or twisted cubics, we show that there are two chambers in the stability manifold where the moduli space is given by a smooth projective irreducible variety, respectively, the Hilbert scheme. In the case of twisted cubics, we compute all walls and moduli spaces on a path between those two chambers. This allows us to give a new proof of the global structure of the main component, originally due to Ellingsrud, Piene, and Strømme. In between slope stability and Bridgeland stability there is the notion of tilt stability that is defined similarly to Bridgeland stability on surfaces. Beyond just $\mathbb {P}^3$, we develop tools to use computations in tilt stability to compute wall crossings in Bridgeland stability.
References
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- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI https://doi.org/10.1093/qmath/45.4.515
- Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 0302647
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- Emanuele Macrì, A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space, Algebra Number Theory 8 (2014), no. 1, 173–190. MR 3207582, DOI https://doi.org/10.2140/ant.2014.8.173
- Antony Maciocia and Ciaran Meachan, Rank 1 Bridgeland stable moduli spaces on a principally polarized abelian surface, Int. Math. Res. Not. IMRN 9 (2013), 2054–2077. MR 3053413, DOI https://doi.org/10.1093/imrn/rns107
- B. Moishezon, On n-dimensional compact complex varieties with n algebraic independent meromorphic functions, Trans. Amer. Math. Soc., 63, 1967, pp. 51–177.
- Antony Maciocia and Dulip Piyaratne, Fourier-Mukai transforms and Bridgeland stability conditions on abelian threefolds, Algebr. Geom. 2 (2015), no. 3, 270–297. MR 3370123, DOI https://doi.org/10.14231/AG-2015-012
- Antony Maciocia and Dulip Piyaratne, Fourier-Mukai transforms and Bridgeland stability conditions on abelian threefolds II, Internat. J. Math. 27 (2016), no. 1, 1650007, 27. MR 3454685, DOI https://doi.org/10.1142/S0129167X16500075
- David Mumford, Further pathologies in algebraic geometry, Amer. J. Math. 84 (1962), 642–648. MR 148670, DOI https://doi.org/10.2307/2372870
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- Benjamin Schmidt, A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold, Bull. Lond. Math. Soc. 46 (2014), no. 5, 915–923. MR 3262194, DOI https://doi.org/10.1112/blms/bdu048
- Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076
- 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
- M. Woolf, Nef and effective cones on the moduli space of torsion sheaves on the projective plane, arXiv:1305.1465v2, 2013.
- Bingyu Xia, Hilbert scheme of twisted cubics as a simple wall-crossing, Trans. Amer. Math. Soc. 370 (2018), no. 8, 5535–5559. MR 3803142, DOI https://doi.org/10.1090/tran/7150
- Shintarou Yanagida and K\B{o}ta Yoshioka, Bridgeland’s stabilities on abelian surfaces, Math. Z. 276 (2014), no. 1-2, 571–610. MR 3150219, DOI https://doi.org/10.1007/s00209-013-1214-1
References
- Daniele Arcara and Aaron Bertram, Bridgeland-stable moduli spaces for $K$-trivial surfaces, with with an appendix by Max Lieblich, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 1–38. MR 2998828, DOI https://doi.org/10.4171/JEMS/354
- Daniele Arcara, Aaron Bertram, Izzet Coskun, and Jack Huizenga, The minimal model program for the Hilbert scheme of points on $\mathbb {P}^2$ and Bridgeland stability, Adv. Math. 235 (2013), 580–626. MR 3010070, DOI https://doi.org/10.1016/j.aim.2012.11.018
- 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
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- 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
- Arend Bayer and Emanuele Macrì, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math. 198 (2014), no. 3, 505–590. MR 3279532, DOI https://doi.org/10.1007/s00222-014-0501-8
- Arend Bayer, Emanuele Macrì, and Paolo Stellari, The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds, Invent. Math. 206 (2016), no. 3, 869–933. MR 3573975, DOI https://doi.org/10.1007/s00222-016-0665-5
- Arend Bayer, Emanuele Macrì, and Yukinobu Toda, Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities, J. Algebraic Geom. 23 (2014), no. 1, 117–163. MR 3121850, DOI https://doi.org/10.1090/S1056-3911-2013-00617-7
- 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
- A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23–42. MR 992977, DOI https://doi.org/10.1070/IM1990v034n01ABEH000583
- 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
- Izzet Coskun, Jack Huizenga, and Matthew Woolf, The effective cone of the moduli space of sheaves on the plane, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 5, 1421–1467. MR 3635357, DOI https://doi.org/10.4171/JEMS/696
- Geir Ellingsrud, Ragni Piene, and Stein Arild Strømme, On the variety of nets of quadrics defining twisted cubics, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 84–96. MR 908709, DOI https://doi.org/10.1007/BFb0078179
- D. Faenzi, A one-day tour of representations and invariants of quivers, Rend. Semin. Mat. Univ. Politec. Torino 71 (2013), no. 1, 3–34. MR 3345057
- Robin Hartshorne, Connectedness of the Hilbert scheme, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 5–48. MR 0213368
- Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077, DOI https://doi.org/10.1007/BF01467074
- G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248. MR 364254, DOI https://doi.org/10.1007/BF01357141
- 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
- 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, DOI https://doi.org/10.1215/kjm/1250283873
- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI https://doi.org/10.1093/qmath/45.4.515
- Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 0302647
- M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435v1, 2008.
- Max Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), no. 1, 175–206. MR 2177199, DOI https://doi.org/10.1090/S1056-3911-05-00418-2
- Jason Lo and Yogesh More, Some examples of tilt-stable objects on threefolds, Comm. Algebra 44 (2016), no. 3, 1280–1301. MR 3463144, DOI https://doi.org/10.1080/00927872.2015.1027348
- Chunyi Li and Xiaolei Zhao, The minimal model program for deformations of Hilbert schemes of points on the projective plane, Algebr. Geom. 5 (2018), no. 3, 328–358. MR 3800356, DOI https://doi.org/10.14231/AG-2018-010
- Antony Maciocia, Computing the walls associated to Bridgeland stability conditions on projective surfaces, Asian J. Math. 18 (2014), no. 2, 263–279. MR 3217637, DOI https://doi.org/10.4310/AJM.2014.v18.n2.a5
- Emanuele Macrì, A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space, Algebra Number Theory 8 (2014), no. 1, 173–190. MR 3207582, DOI https://doi.org/10.2140/ant.2014.8.173
- Antony Maciocia and Ciaran Meachan, Rank 1 Bridgeland stable moduli spaces on a principally polarized abelian surface, Int. Math. Res. Not. IMRN 9 (2013), 2054–2077. MR 3053413, DOI https://doi.org/10.1093/imrn/rns107
- B. Moishezon, On n-dimensional compact complex varieties with n algebraic independent meromorphic functions, Trans. Amer. Math. Soc., 63, 1967, pp. 51–177.
- Antony Maciocia and Dulip Piyaratne, Fourier-Mukai transforms and Bridgeland stability conditions on abelian threefolds, Algebr. Geom. 2 (2015), no. 3, 270–297. MR 3370123, DOI https://doi.org/10.14231/AG-2015-012
- Antony Maciocia and Dulip Piyaratne, Fourier-Mukai transforms and Bridgeland stability conditions on abelian threefolds II, Internat. J. Math. 27 (2016), no. 1, 1650007, 27. MR 3454685, DOI https://doi.org/10.1142/S0129167X16500075
- David Mumford, Further pathologies in algebraic geometry, Amer. J. Math. 84 (1962), 642–648. MR 148670, DOI https://doi.org/10.2307/2372870
- Howard Nuer, Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface, Proc. Lond. Math. Soc. (3) 113 (2016), no. 3, 345–386. MR 3551850, DOI https://doi.org/10.1112/plms/pdw033
- Ragni Piene and Michael Schlessinger, On the Hilbert scheme compactification of the space of twisted cubics, Amer. J. Math. 107 (1985), no. 4, 761–774. MR 796901, DOI https://doi.org/10.2307/2374355
- Dulip Piyaratne and Yukinobu Toda, Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas invariants, J. Reine Angew. Math. 747 (2019), 175–219. MR 3905133, DOI https://doi.org/10.1515/crelle-2016-0006
- W. A. Stein et al., Sage Mathematics Software (Version 6.6), The Sage Development Team, 2015, http://www.sagemath.org.
- Benjamin Schmidt, A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold, Bull. Lond. Math. Soc. 46 (2014), no. 5, 915–923. MR 3262194, DOI https://doi.org/10.1112/blms/bdu048
- Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076
- 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
- M. Woolf, Nef and effective cones on the moduli space of torsion sheaves on the projective plane, arXiv:1305.1465v2, 2013.
- Bingyu Xia, Hilbert scheme of twisted cubics as a simple wall-crossing, Trans. Amer. Math. Soc. 370 (2018), no. 8, 5535–5559. MR 3803142, DOI https://doi.org/10.1090/tran/7150
- Shintarou Yanagida and Kōta Yoshioka, Bridgeland’s stabilities on abelian surfaces, Math. Z. 276 (2014), no. 1-2, 571–610. MR 3150219, DOI https://doi.org/10.1007/s00209-013-1214-1
Additional Information
Benjamin Schmidt
Affiliation:
Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, Texas 78712
Address at time of publication:
Gottfried Wilhelm Leibniz Universität Hannover, Institut für Algebraische Geometrie, Welfengarten 1, 30167 Hannover, Germany
MR Author ID:
1079080
Email:
bschmidt@math.uni-hannover.de
Received by editor(s):
September 19, 2016
Received by editor(s) in revised form:
July 20, 2018
Published electronically:
November 15, 2019
Additional Notes:
This research was partially supported by NSF grants DMS-1160466 and DMS-1523496 (PI Emanuele Macrì) and a presidential fellowship of the Ohio State University.
Article copyright:
© Copyright 2019
University Press, Inc.