Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions
Authors:
Georg Oberdieck, Dulip Piyaratne and Yukinobu Toda
Journal:
J. Algebraic Geom. 31 (2022), 13-73
DOI:
https://doi.org/10.1090/jag/788
Published electronically:
September 14, 2021
MathSciNet review:
4372406
Full-text PDF
Abstract |
References |
Additional Information
Abstract:
We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author.
For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.
References
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- Tom Bridgeland, Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), no. 4, 969–998. MR 2813335, DOI 10.1090/S0894-0347-2011-00701-7
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- 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 10.4310/AJM.2014.v18.n2.a5
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- Shigeru Mukai, Semi-homogeneous vector bundles on an Abelian variety, J. Math. Kyoto Univ. 18 (1978), no. 2, 239–272. MR 498572, DOI 10.1215/kjm/1250522574
- Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081, DOI 10.1017/S002776300001922X
- Georg Oberdieck, On reduced stable pair invariants, Math. Z. 289 (2018), no. 1-2, 323–353. MR 3803792, DOI 10.1007/s00209-017-1953-5
- Georg Oberdieck and Junliang Shen, Reduced Donaldson-Thomas invariants and the ring of dual numbers, Proc. Lond. Math. Soc. (3) 118 (2019), no. 1, 191–220. MR 3898990, DOI 10.1112/plms.12178
- Georg Oberdieck and Junliang Shen, Curve counting on elliptic Calabi-Yau threefolds via derived categories, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 3, 967–1002. MR 4055994, DOI 10.4171/jems/938
- D. O. Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 131–158 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 3, 569–594. MR 1921811, DOI 10.1070/IM2002v066n03ABEH000389
- Dulip Piyaratne, Stability conditions under the Fourier-Mukai transforms on abelian 3-folds, Q. J. Math. 70 (2019), no. 1, 225–288. MR 3927850, DOI 10.1093/qmath/hay036
- 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 10.1515/crelle-2016-0006
- Matthieu Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), no. 1, 209–236. MR 2125542, DOI 10.1307/mmj/1114021093
- SageMath, the Sage Mathematics Software System (Version 7.4), The Sage Developers, 2016, http://www.sagemath.org.
- Ashoke Sen, $\scr N=8$ dyon partition function and walls of marginal stability, J. High Energy Phys. 7 (2008), 118, 18. MR 2430066, DOI 10.1088/1126-6708/2008/07/118
- Junliang Shen, The Euler characteristics of generalized Kummer schemes, Math. Z. 281 (2015), no. 3-4, 1183–1189. MR 3421659, DOI 10.1007/s00209-015-1526-4
- Yukinobu Toda, Generating functions of stable pair invariants via wall-crossings in derived categories, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) Adv. Stud. Pure Math., vol. 59, Math. Soc. Japan, Tokyo, 2010, pp. 389–434. MR 2683216, DOI 10.2969/aspm/05910389
- Yukinobu Toda, Stable pairs on local K3 surfaces, J. Differential Geom. 92 (2012), no. 2, 285–371. MR 2998674
- Yukinobu Toda, Hall algebras in the derived category and higher-rank DT invariants, Algebr. Geom. 7 (2020), no. 3, 240–262. MR 4087861, DOI 10.14231/ag-2020-008
References
- J. Alper, D. Halpern-Leistner, and J. Heinloth, Existence of moduli space for algebraic stacks, arXiv:1812.01128 (2018).
- Jarod Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402 (English, with English and French summaries). MR 3237451
- 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 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 10.1090/S1056-3911-2013-00617-7
- T. Beckmann and G. Oberdieck, On equivariant derived categories, arXiv:2006.13626 (2020).
- Kai Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338. MR 2600874, DOI 10.4007/annals.2009.170.1307
- C. Birkenhake and H. Lange, The dual polarization of an abelian variety, Arch. Math. (Basel) 73 (1999), no. 5, 380–389. MR 1712138, DOI 10.1007/s000130050412
- Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI 10.4007/annals.2007.166.317
- Tom Bridgeland, Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), no. 4, 969–998. MR 2813335, DOI 10.1090/S0894-0347-2011-00701-7
- Jim Bryan, Georg Oberdieck, Rahul Pandharipande, and Qizheng Yin, Curve counting on abelian surfaces and threefolds, Algebr. Geom. 5 (2018), no. 4, 398–463. MR 3813750, DOI 10.14231/ag-2018-012
- Martin G. Gulbrandsen, Donaldson-Thomas invariants for complexes on abelian threefolds, Math. Z. 273 (2013), no. 1-2, 219–236. MR 3010158, DOI 10.1007/s00209-012-1002-3
- D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106, DOI 10.1093/acprof:oso/9780199296866.001.0001
- Jun-ichi Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997–1028. MR 277558, DOI 10.2307/2373406
- Dominic Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153–219. MR 2354988, DOI 10.1016/j.aim.2007.04.002
- Dominic Joyce, Motivic invariants of Artin stacks and ‘stack functions’, Q. J. Math. 58 (2007), no. 3, 345–392. MR 2354923, DOI 10.1093/qmath/ham019
- Dominic Joyce and Yinan Song, A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), no. 1020, iv+199. MR 2951762, DOI 10.1090/S0065-9266-2011-00630-1
- Jason Lo and Zhenbo Qin, Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces, Asian J. Math. 18 (2014), no. 2, 321–344. MR 3217639, DOI 10.4310/AJM.2014.v18.n2.a7
- 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 10.4310/AJM.2014.v18.n2.a5
- 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 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 pp.
- D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), no. 5, 1263–1285. MR 2264664, DOI 10.1112/S0010437X06002302
- Davesh Maulik and Richard P. Thomas, Sheaf counting on local K3 surfaces, Pure Appl. Math. Q. 14 (2018), no. 3-4, 419–441. MR 4047404, DOI 10.4310/PAMQ.2018.v14.n3.a1
- S. Mukai, Abelian variety and spin representation, Proceedings of symposium Hodge theory and algebraic geometry (Sapporo, 1994), pp. 110–135 (in Japanese, English translation: Univ. of Warwick, preprint, 1998, www.kurims.kyoto-u.ac.jp/~mukai).
- Shigeru Mukai, Semi-homogeneous vector bundles on an Abelian variety, J. Math. Kyoto Univ. 18 (1978), no. 2, 239–272. MR 498572, DOI 10.1215/kjm/1250522574
- Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081
- Georg Oberdieck, On reduced stable pair invariants, Math. Z. 289 (2018), no. 1-2, 323–353. MR 3803792, DOI 10.1007/s00209-017-1953-5
- Georg Oberdieck and Junliang Shen, Reduced Donaldson-Thomas invariants and the ring of dual numbers, Proc. Lond. Math. Soc. (3) 118 (2019), no. 1, 191–220. MR 3898990, DOI 10.1112/plms.12178
- Georg Oberdieck and Junliang Shen, Curve counting on elliptic Calabi-Yau threefolds via derived categories, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 3, 967–1002. MR 4055994, DOI 10.4171/jems/938
- D. O. Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 131–158 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 3, 569–594. MR 1921811, DOI 10.1070/IM2002v066n03ABEH000389
- Dulip Piyaratne, Stability conditions under the Fourier-Mukai transforms on abelian 3-folds, Q. J. Math. 70 (2019), no. 1, 225–288. MR 3927850, DOI 10.1093/qmath/hay036
- 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 10.1515/crelle-2016-0006
- Matthieu Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), no. 1, 209–236. MR 2125542, DOI 10.1307/mmj/1114021093
- SageMath, the Sage Mathematics Software System (Version 7.4), The Sage Developers, 2016, http://www.sagemath.org.
- Ashoke Sen, $\mathcal {N}=8$ dyon partition function and walls of marginal stability, J. High Energy Phys. 7 (2008), 118, 18. MR 2430066, DOI 10.1088/1126-6708/2008/07/118
- Junliang Shen, The Euler characteristics of generalized Kummer schemes, Math. Z. 281 (2015), no. 3-4, 1183–1189. MR 3421659, DOI 10.1007/s00209-015-1526-4
- Yukinobu Toda, Generating functions of stable pair invariants via wall-crossings in derived categories, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) Adv. Stud. Pure Math., vol. 59, Math. Soc. Japan, Tokyo, 2010, pp. 389–434. MR 2683216, DOI 10.2969/aspm/05910389
- Yukinobu Toda, Stable pairs on local K3 surfaces, J. Differential Geom. 92 (2012), no. 2, 285–371. MR 2998674
- Yukinobu Toda, Hall algebras in the derived category and higher-rank DT invariants, Algebr. Geom. 7 (2020), no. 3, 240–262. MR 4087861, DOI 10.14231/ag-2020-008
Additional Information
Georg Oberdieck
Affiliation:
Mathematisches Institut, Universität Bonn, Germany
MR Author ID:
1175196
Email:
georgo@math.uni-bonn.de
Dulip Piyaratne
Affiliation:
Department of Mathematics, Xiamen University of Malaysia, Malaysia
MR Author ID:
1116811
ORCID:
0000-0002-6945-8664
Email:
piyaratne@xmu.edu.my
Yukinobu Toda
Affiliation:
Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo (WPI), Japan
MR Author ID:
792353
Email:
yukinobu.toda@ipmu.jp
Received by editor(s):
August 22, 2018
Received by editor(s) in revised form:
May 25, 2021
Published electronically:
September 14, 2021
Additional Notes:
The first author was supported by the National Science Foundation Grant DMS-1440140 while in residence at MSRI, Berkeley. The second author was supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan. The third author was supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and Grant-in Aid for Scientific Research grant (No. 26287002) from MEXT, Japan
Article copyright:
© Copyright 2021
University Press, Inc.