Duke Mathematical Journal

A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory

Johannes Nicaise and Sam Payne

Full-text: Access denied (subscription has expired)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semialgebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson–Thomas theory.

Article information

Duke Math. J., Volume 168, Number 10 (2019), 1843-1886.

Received: 5 September 2017
Revised: 19 November 2018
First available in Project Euclid: 7 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14E18: Arcs and motivic integration 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]

motivic integration nearby cycles Donaldson–Thomas theory tropical geometry


Nicaise, Johannes; Payne, Sam. A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory. Duke Math. J. 168 (2019), no. 10, 1843--1886. doi:10.1215/00127094-2019-0003. https://projecteuclid.org/euclid.dmj/1559894420

Export citation


  • [1] K. Behrend, J. Bryan, and B. Szendrői, Motivic degree zero Donaldson-Thomas invariants, Invent. Math. 192 (2013), no. 1, 111–160.
  • [2] F. Bittner, On motivic zeta functions and the motivic nearby fiber, Math. Z. 249 (2005), no. 1, 63–83.
  • [3] L. Borisov, The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom. 27 (2018), no. 2, 203–209.
  • [4] E. Bultot and J. Nicaise, Computing motivic zeta functions on log smooth models, preprint, arXiv:1610.00742v3 [math.AG].
  • [5] B. Davison and S. Meinhardt, Motivic Donaldson-Thomas invariants for the one-loop quiver with potential, Geom. Topol. 19 (2015), no. 5, 2535–2555.
  • [6] B. Davison and S. Meinhardt, The motivic Donaldson-Thomas invariants of $(-2)$-curves, Algebra Number Theory 11 (2017), no. 6, 1243–1286.
  • [7] J. Denef and F. Loeser, “Geometry on arc spaces of algebraic varieties” in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, Basel, 2001, 327–348.
  • [8] W. Gubler, “A guide to tropicalizations” in Algebraic and Combinatorial Aspects of Tropical Geometry (Castro Urdiales, 2011), Contemp. Math. 589, Amer. Math. Soc., Providence, 2013, 125–189.
  • [9] G. Guibert, F. Loeser, and M. Merle, Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink, Duke Math. J. 132 (2006), no. 3, 409–457.
  • [10] E. Hrushovski and D. Kazhdan, “Integration in valued fields” in Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser, Boston, 2006, 261–405.
  • [11] E. Hrushovski and F. Loeser, Monodromy and the Lefschetz fixed point formula, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 313–349.
  • [12] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435v1 [math.AG].
  • [13] Q. T. Lê, Proofs of the integral identity conjecture over algebraically closed fields, Duke Math. J. 164 (2015), no. 1, 157–194.
  • [14] M. Luxton and Z. Qu, Some results on tropical compactifications, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4853–4876.
  • [15] J. Nicaise, Geometric criteria for tame ramification, Math. Z. 273 (2013), nos. 3–4, 839–868.
  • [16] J. Nicaise, “Geometric invariants for non-archimedean semialgebraic sets” in Algebraic Geometry (Salt Lake City, 2015), Proc. Sympos. Pure Math. 97, Amer. Math. Soc., Providence, 2018, 389–404.
  • [17] J. Nicaise, S. Payne, and F. Schroeter, Tropical refined curve counting via motivic integration, Geom. Topol. 22 (2018), no. 6, 3175–3234.
  • [18] J. Nicaise and J. Sebag, “The Grothendieck ring of varieties” in Motivic Integration and Its Interactions with Model Theory and Non-Archimedean Geometry, Vol. I, London Math. Soc. Lecture Note Ser. 383, Cambridge Univ. Press, Cambridge, 2011, 145–188.
  • [19] J. Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), no. 4, 1087–1104.