Geometry & Topology

Zero dimensional Donaldson–Thomas invariants of threefolds

Jun Li

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Using a homotopy approach, we prove in this paper a conjecture of Maulik, Nekrasov, Okounkov and Pandharipande on the dimension zero Donaldson–Thomas invariants of all smooth complex threefolds.

Article information

Geom. Topol., Volume 10, Number 4 (2006), 2117-2171.

Received: 27 April 2006
Accepted: 10 October 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]

moduli space Hilbert schemes virtual cycle


Li, Jun. Zero dimensional Donaldson–Thomas invariants of threefolds. Geom. Topol. 10 (2006), no. 4, 2117--2171. doi:10.2140/gt.2006.10.2117.

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  • I V Artamkin, On the deformation of sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988) 660–665, 672 (Russian) English translation: Math. USSR. Izvestiya, 32 (1989) 663–668
  • K Behrend, Donaldson-Thomas invariants via microlocal geometry
  • K Behrend, B Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds
  • K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45–88
  • S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: “The geometric universe science, geometry, and the work of Roger Penrose (Oxford, 1996)”, (S S Huggett, et al, editors), Oxford Univ. Press (1998) 31–47
  • M Levine, R Pandharipande, Algebraic Cobordism revisited
  • J Li, G Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119–174
  • J Li, G Tian, Comparison of algebraic and symplectic Gromov-Witten invariants, Asian J. Math. 3 (1999) 689–728
  • M Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ. 18 (1978) 557–614
  • D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, I
  • D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, I
  • S Mukai, Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math. 77 (1984) 101–116
  • R P Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000) 367–438