Avec cedram.org
Confluentes Mathematici
Rechercher un article
Recherche sur le site
Table des matières de ce fascicule | Article précédent
Sven Meinhardt
An Introduction to (Motivic) Donaldson-Thomas Theory
Confluentes Mathematici, 9 no. 2 (2017), p. 101-158, doi: 10.5802/cml.43
Article PDF
Class. Math.: 14N35, 14D23, 16G20, 32S60, 55N33
Mots clés: moduli stacks, Grothendieck groups of varieties, Donaldson-Thomas invariants, quiver representations

Résumé - Abstract

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.


[1] K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170(3):1307–1338, 2009. arXiv:math.AG/0507523.
[2] K. Behrend, J. Byan, and B. Szendrői, Motivic degree zero Donaldson-Thomas invariants, Invent. Math. 192:111–160, 2013. arXiv:0909.5088.
[3] F. Bittner. The universal euler characteristic for varieties of characteristic zero. Comp. Math., 140:1011–1032, 2004.
[4] B. Davison and S. Meinhardt, The motivic Donaldson-Thomas invariants of $(-2)$ curves, to appear in Alg. Number Th., 2012. arXiv:1208.2462.
[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.
[7] B. Davison and S. Meinhardt, Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras, 2016. arXiv:1601.02479.
[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.
[9] W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. J. 27:91–94, 1960.
[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.
[11] D. Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203:194–255, 2006. arXiv:math.AG/0312190.
[12] D. Joyce Constructable functions on Artin stacks, J. Lon. Math. Soc. 74(3):583–606, 2006. arXiv:math.AG/0403305.
[13] D. Joyce, Configurations in abelian categories. II. Ringel–Hall algebras, Adv. Math. 210:635–706, 2007. arXiv:math.AG/0503029.
[14] D. Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215:153–219, 2007. arXiv:math.AG/0410267.
[15] D. Joyce, Motivic invariants of Artin stacks and ‘stack functions’, Quart. J. Math. 58(3):345–392, 2007. arXiv:math.AG/0509722.
[16] D. Joyce, Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217:125–204, 2008. arXiv:math.AG/0503029.
[17] D. Joyce and Y. Song, A theory of generalized Donaldson-Thomas invariants, Mem.Amer. Math. Soc. 217(1020), 2012. arXiv:math.AG/08105645.
[18] A. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. 45(4):515–530, 1994.
[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.
[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.
[22] E. Looijenga, Motivic measures, Astérisque 276:267–297, 2002. Séminaire Bourbaki, 1999/ 2000.
[23] R. D. MacPherson, Chern Classes for Singular Algebraic Variaties, Ann. Math. 100(2):423–432, 1974.
[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.
[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.
[28] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152(2):349–368, 2003.
[29] M. Reineke, Moduli of representations of quivers, Proceedings of the ICRA XII conference, Torun, 2007. arXiv:0802.2147.
[30] M. Reineke, Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9(3):653–667, 2010. arXiv:0804.3214.
[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.
[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.
[33] M. Reineke and S. Schröer, Brauer groups for quiver moduli, 2014. arxiv:1410.0466.
[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.
[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.
eISSN : 1793-7434