Algebra & Number Theory

The motivic Donaldson–Thomas invariants of ($-$2)-curves

Ben Davison and Sven Meinhardt

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We calculate the motivic Donaldson–Thomas invariants for (2)-curves arising from 3-fold flopping contractions in the minimal model program. We translate this geometric situation into the machinery developed by Kontsevich and Soibelman, and using the results and framework developed earlier by the authors we describe the monodromy on these invariants. In particular, in contrast to all existing known Donaldson–Thomas invariants for small resolutions of Gorenstein singularities these monodromy actions are nontrivial.

Article information

Algebra Number Theory, Volume 11, Number 6 (2017), 1243-1286.

Received: 6 February 2016
Revised: 23 November 2016
Accepted: 1 February 2017
First available in Project Euclid: 16 November 2017

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Donaldson–Thomas theory minus two curves motivic invariants


Davison, Ben; Meinhardt, Sven. The motivic Donaldson–Thomas invariants of ($-$2)-curves. Algebra Number Theory 11 (2017), no. 6, 1243--1286. doi:10.2140/ant.2017.11.1243.

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  • P. S. Aspinwall and S. Katz, “Computation of superpotentials for D-branes”, Comm. Math. Phys. 264:1 (2006), 227–253.
  • K. Behrend and B. Fantechi, “The intrinsic normal cone”, Invent. Math. 128:1 (1997), 45–88.
  • K. Behrend, J. Bryan, and B. Szendrői, “Motivic degree zero Donaldson–Thomas invariants”, Invent. Math. 192:1 (2013), 111–160.
  • M. van den Bergh, “Non-commutative crepant resolutions”, pp. 749–770 in The legacy of Niels Henrik Abel, edited by O. A. Laudal and R. Piene, Springer, Berlin, 2004.
  • T. Bridgeland, “An introduction to motivic Hall algebras”, Adv. Math. 229:1 (2012), 102–138.
  • J. Bryan, S. Katz, and N. C. Leung, “Multiple covers and the integrality conjecture for rational curves in Calabi–Yau threefolds”, J. Alg. Geom. 10:3 (2001), 549–568.
  • H. Clemens, J. Kollár, and S. Mori, Higher-dimensional complex geometry, Astérisque 166, Société Mathématique de France, Paris, 1988.
  • B. Davison, “Invariance of orientation data for ind-constructible Calabi–Yau $A_{\infty}$ categories under derived equivalence”, preprint, 2010.
  • B. Davison and S. Meinhardt, “Motivic Donaldson–Thomas invariants for the one-loop quiver with potential”, Geom. Topol. 19:5 (2015), 2535–2555.
  • J. Denef and F. Loeser, “Motivic Igusa zeta functions”, J. Alg. Geom. 7:3 (1998), 505–537.
  • A. Dimca and B. Szendrői, “The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on $\mathbb{C}^3$”, Math. Res. Lett. 16:6 (2009), 1037–1055.
  • T. Ekedahl, “The Grothendieck group of algebraic stacks”, preprint, 2009.
  • W. Feit and N. J. Fine, “Pairs of commuting matrices over a finite field”, Duke Math. J 27 (1960), 91–94.
  • V. Ginzburg, “Calabi–Yau algebras”, preprint, 2006.
  • G. Guibert, F. Loeser, and M. Merle, “Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink”, Duke Math. J. 132:3 (2006), 409–457.
  • S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “A power structure over the Grothendieck ring of varieties”, Math. Res. Lett. 11:1 (2004), 49–57.
  • \relax minus 2pt D. Joyce, “Configurations in abelian categories, I: Basic properties and moduli stacks”, Adv. Math. 203:1 (2006), 194–255.
  • D. Joyce, “Constructible functions on Artin stacks”, J. London Math. Soc. $(2)$ 74:3 (2006), 583–606.
  • D. Joyce, “Configurations in abelian categories, II: Ringel–Hall algebras”, Adv. Math. 210:2 (2007), 635–706.
  • D. Joyce, “Configurations in abelian categories, III: Stability conditions and identities”, Adv. Math. 215:1 (2007), 153–219.
  • D. Joyce, “Holomorphic generating functions for invariants counting coherent sheaves on Calabi–Yau 3-folds”, Geom. Topol. 11 (2007), 667–725.
  • D. Joyce, “Motivic invariants of Artin stacks and `stack functions”', Q. J. Math. 58:3 (2007), 345–392.
  • D. Joyce, “Configurations in abelian categories, IV: Invariants and changing stability conditions”, Adv. Math. 217:1 (2008), 125–204.
  • D. Joyce and Y. Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 1020, American Mathematical Society, Providence, RI, 2012.
  • H. Kajiura, “Noncommutative homotopy algebras associated with open strings”, Rev. Math. Phys. 19:1 (2007), 1–99.
  • S. Katz and D. R. Morrison, “Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups”, J. Alg. Geom. 1:3 (1992), 449–530.
  • B. Keller, “Introduction to $A$-infinity algebras and modules”, Homology Homotopy Appl. 3:1 (2001), 1–35.
  • J. Kollár, “Flops”, Nagoya Math. J. 113 (1989), 15–36.
  • M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson–Thomas invariants and cluster transformations”, preprint, 2008.
  • M. Kontsevich and Y. Soibelman, “Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry”, pp. 153–219 in Homological mirror symmetry, edited by A. Kapustin et al., Lecture Notes in Phys. 757, Springer, Berlin, 2009.
  • M. Kontsevich and Y. Soibelman, “Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants”, Commun. Number Theory Phys. 5:2 (2011), 231–352.
  • H. B. Laufer, “On ${\mathbb C}P^{1}$ as an exceptional set”, pp. 261–275 in Recent developments in several complex variables (Princeton, 1979), edited by J. E. Fornaess, Ann. of Math. Stud. 100, Princeton Univ. Press, 1981.
  • Q. T. Lê, “Proofs of the integral identity conjecture over algebraically closed fields”, Duke Math. J. 164:1 (2015), 157–194.
  • K. Lefèvre-Hasegawa, Sur les $A_{\infty}$-catégories, Ph.D. thesis, Université Paris Diderot, 2003.
  • E. Looijenga, “Motivic measures”, pp. [exposé] 874, pp. 267–297 in Séminaire Bourbaki, 1999/2000, Astérisque 276, Soc. Mat. de France, Paris, 2002.
  • D. M. Lu, J. H. Palmieri, Q. S. Wu, and J. J. Zhang, “Koszul equivalences in $A_\infty$-algebras”, New York J. Math. 14 (2008), 325–378.
  • D. Maulik, “Motivic residues and Donaldson–Thomas theory”, preprint, 2013.
  • A. Morrison and K. Nagao, “Motivic Donaldson–Thomas invariants of small crepant resolutions”, Algebra Number Theory 9:4 (2015), 767–813.
  • 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 (2012), 2065–2093.
  • K. Nagao and H. Nakajima, “Counting invariant of perverse coherent sheaves and its wall-crossing”, Int. Math. Res. Not. 2011:17 (2011), 3885–3938.
  • M. Reid, “Minimal models of canonical $3$-folds”, pp. 131–180 in Algebraic varieties and analytic varieties (Tokyo, 1981), edited by S. Iitaka, Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983.
  • B. Szendrői, “Non-commutative Donaldson–Thomas invariants and the conifold”, Geom. Topol. 12:2 (2008), 1171–1202.
  • R. P. Thomas, “A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on $K3$ fibrations”, J. Differential Geom. 54:2 (2000), 367–438.
  • Y. Toda, “On a certain generalization of spherical twists”, Bull. Soc. Math. France 135:1 (2007), 119–134.
  • B. Young, “Computing a pyramid partition generating function with dimer shuffling”, J. Combin. Theory Ser. A 116:2 (2009), 334–350.