An Introduction to (Motivic) Donaldson-Thomas Theory
Confluentes Mathematici, Tome 9 (2017) no. 2, pp. 101-158.

The aim of the paper is to provide a rather gentle introduction into Donaldson-Thomas theory using quivers with potential. The reader should be familiar with some basic knowledge in algebraic or complex geometry. The text contains many examples and exercises to support the process of understanding the main concepts and ideas.

Reçu le : 2016-01-15
Révisé le : 2017-07-29
Accepté le : 2017-11-16
Publié le : 2017-12-14
DOI : https://doi.org/10.5802/cml.43
Classification : 14N35,  14D23,  16G20,  32S60,  55N33
Mots clés: moduli stacks, Grothendieck groups of varieties, Donaldson-Thomas invariants, quiver representations
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     author = {Sven Meinhardt},
     title = {An Introduction to (Motivic) Donaldson-Thomas Theory},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {9},
     number = {2},
     year = {2017},
     pages = {101-158},
     doi = {10.5802/cml.43},
     zbl = {1400.14142},
     mrnumber = {3745163},
     language = {en},
     url = {cml.centre-mersenne.org/item/CML_2017__9_2_101_0/}
}
Sven Meinhardt. An Introduction to (Motivic) Donaldson-Thomas Theory. Confluentes Mathematici, Tome 9 (2017) no. 2, pp. 101-158. doi : 10.5802/cml.43. https://cml.centre-mersenne.org/item/CML_2017__9_2_101_0/

[1] K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170(3):1307–1338, 2009. arXiv:math.AG/0507523. | Article | MR 2600874 | Zbl 1191.14050

[2] K. Behrend, J. Byan, and B. Szendrői, Motivic degree zero Donaldson-Thomas invariants, Invent. Math. 192:111–160, 2013. arXiv:0909.5088. | Article | MR 3032328 | Zbl 1267.14008

[3] F. Bittner. The universal euler characteristic for varieties of characteristic zero. Comp. Math., 140:1011–1032, 2004. | Article | MR 2059227 | Zbl 1086.14016

[4] B. Davison and S. Meinhardt, The motivic Donaldson-Thomas invariants of (-2) curves, to appear in Alg. Number Th., 2012. arXiv:1208.2462. | Article | MR 3687097 | Zbl 1409.14092

[5] B. Davison and S. Meinhardt, Donaldson-Thomas theory for categories of homological dimension one with potential, 2015. arXiv:1512.08898.

[6] B. Davison and S. Meinhardt, Motivic DT-invariants for the one loop quiver with potential, Geometry and Topology 19:2535–2555, 2015. https://doi.org/10.2140/gt.2015.19.2535. | Article | MR 3416109 | Zbl 1430.14105

[7] B. Davison and S. Meinhardt, Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras, 2016. arXiv:1601.02479. | Article

[8] J. Denef and F. Loeser, Geometry on arc spaces of algebraic varieties, In Europ. Cong. Math., Vol. I (Barcelona, 2000), Progr. Math. 201, pages 327–348. Birkhäuser. | Article | Zbl 1079.14003

[9] W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. J. 27:91–94, 1960. | Article | MR 109810 | Zbl 0097.00702

[10] S. M. Guseine-Zade, I. Luengo, and A. Melle-Hernández, A power structure over the Grothendieck ring of varieties, Math. Res. Lett. 11(1):49–57, 2004. arXiv:math.AG/0206279. | Article | MR 2046199 | Zbl 1063.14026

[11] D. Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203:194–255, 2006. arXiv:math.AG/0312190. | Article | MR 2231046 | Zbl 1102.14009

[12] D. Joyce Constructable functions on Artin stacks, J. Lon. Math. Soc. 74(3):583–606, 2006. arXiv:math.AG/0403305. | Article | MR 2286434 | Zbl 1112.14004

[13] D. Joyce, Configurations in abelian categories. II. Ringel–Hall algebras, Adv. Math. 210:635–706, 2007. arXiv:math.AG/0503029. | Article | MR 2303235 | Zbl 1119.14005

[14] D. Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215:153–219, 2007. arXiv:math.AG/0410267. | Article | MR 2354988 | Zbl 1134.14007

[15] D. Joyce, Motivic invariants of Artin stacks and ‘stack functions’, Quart. J. Math. 58(3):345–392, 2007. arXiv:math.AG/0509722. | Article | MR 2354923 | Zbl 1131.14005

[16] D. Joyce, Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217:125–204, 2008. arXiv:math.AG/0503029. | Article | MR 2357325 | Zbl 1134.14008

[17] D. Joyce and Y. Song, A theory of generalized Donaldson-Thomas invariants, Mem.Amer. Math. Soc. 217(1020), 2012. arXiv:math.AG/08105645. | Article | MR 2951762 | Zbl 1259.14054

[18] A. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. 45(4):515–530, 1994. | Article | MR 1315461 | Zbl 0837.16005

[19] M. Kontsevich and J. Soibelman, Stability structures, motive Donaldson-Thomas invariants and cluster transformations, 2008. arXiv:math.AG/08112435.

[20] M. Kontsevich and Y. Soibelman, Motivic Donaldson-Thomas invariants: summary of results. In Mirror symmetry and tropical geometry, Contemp. Math. 527, pages 55–89. Amer. Math. Soc., Providence, RI, 2010. | Article | Zbl 1214.14014

[21] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Comm. Number Th. Phys. 5(2):231–252, 2011. arXiv:1006.2706. | Article | MR 2851153 | Zbl 1248.14060

[22] E. Looijenga, Motivic measures, Astérisque 276:267–297, 2002. Séminaire Bourbaki, 1999/ 2000. | Numdam | Zbl 0996.14011

[23] R. D. MacPherson, Chern Classes for Singular Algebraic Variaties, Ann. Math. 100(2):423–432, 1974. | Article | MR 361141 | Zbl 0311.14001

[24] S. Meinhardt, Donaldson–Thomas invariants versus intersection cohomology for categories of homological dimension one, 2015. arXiv:1512.03343.

[25] S. Meinhardt, Moduli spaces of quiver representations and rational points in wall intersections, 2017. In preparation.

[26] S. Meinhardt and M. Reineke, Donaldson-Thomas invariants versus intersection cohomology of quiver moduli, J. Reine Angew. Math. 2017. https://doi.org/10.1515/crelle-2017-0010. | Article | MR 4000572 | Zbl 07105906

[27] A. Morrison, S. Mozgovoy, K. Nagao, and B. Szendrői, Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex, Adv. Math. 230(4–6):2065–2093, 2012. | Article | MR 2927365 | Zbl 1257.14028

[28] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152(2):349–368, 2003. | Article | MR 1974891 | Zbl 1043.17010

[29] M. Reineke, Moduli of representations of quivers, Proceedings of the ICRA XII conference, Torun, 2007. arXiv:0802.2147. | Article | Zbl 1206.16009

[30] M. Reineke, Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9(3):653–667, 2010. arXiv:0804.3214. | Article | MR 2650811 | Zbl 1232.53072

[31] M. Reineke, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, Comp. Math. 147(3):943–964, 2011. arXiv:0903.0261. | Article | MR 2801406 | Zbl 1266.16013

[32] M. Reineke, Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers, Doc. Math. 17:1–22, 2012. arXiv:1102.3978. | Zbl 1280.16018

[33] M. Reineke and S. Schröer, Brauer groups for quiver moduli, 2014. arxiv:1410.0466. | Article | MR 3683503

[34] The Stacks Project Authors, Stacks project, 2017. http://stacks.math.columbia.edu

[35] B. Szendrői, Cohomological Donaldson-Thomas theory, Proc. String-Math. 2014, 2015. arXiv:1503.07349. | Zbl 1378.14059

[36] R. P. Thomas, A holomorphic casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Diff. Geom. 54:367–438, 2000. math.AG/9806111. | Article | MR 1818182 | Zbl 1034.14015