##
**A tropical motivic Fubini theorem with applications to Donaldson-Thomas Theory.**
*(English)*
Zbl 1429.14033

Studying hypersurface singularities over a field \(k\), J. Denef and F. Loeser introduced some important invariants [Prog. Math. 201, 327–348 (2001; Zbl 1079.14003)], such as motivic nearby fiber, motivic vanishing cycles and motivic Milnor fiber. They are defined in the localized Grothendieck ring \(\mathbb{K}^{\hat{\mu}}(Var_k)[\mathbb{L}^{-1}]\) of varieties over \(k\) with an action of the profinite group of roots of unity \(\hat{\mu}\). In the article under review is proposed new approach for their calculation, based on tropical geometry and Hrushovski-Kazhdan motivic integration [E. Hrushovski and D. Kazhdan, Prog. Math. 253, 261–405 (2006; Zbl 1136.03025)], based on model theory of algebraically closed valued fields. The advantage is twofold – these invariants are defined yet in \(\mathbb{K}^{\hat{\mu}}(Var_k)\), and, interpreting the motivic nearby fiber and the motivic Milnor fiber as volumes of semialgebraic sets permits to exploit natural connections with tropical geometry. Moreover, a motivic Fubini theorem gives a new method to compute these volumes. As applications are given new proofs of the conjectures of B. Davison and S. Meinhardt [Geom. Topol. 19, No. 5, 2535–2555 (2015; Zbl 1430.14105)], and the integral identity conjecture of M. Kontsevich and Y. Soibelman [“Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, arXiv:0811.2435], with shorter and more conceptual proofs.

Supposed the field \(k\) is of characteristic \(0\), containing all roots of unity. Let \(R = k[[t]]\), \(K_0 = k((t))\subset K\) be the field of Puiseux series, and pick up \(\bar{K}\) an algebraic closure. We have \(\hat{\mu}\simeq \mathrm{Gal} (K/K_0)\), and let \(\bar{R}\subset\bar{K}\) be the valuation rings with respect to the \(t\)-adic valuation. For \(\mathbb{T}\) a split \(R\)-torus with character lattice \(M\) of finite rank, define the tropicalization map \(\mathrm{trop} :\mathbb{T}(\bar{K})\to \mathrm{Hom}(M, \mathbb{Q})\), \(x\mapsto (m \mapsto \mathrm{val}(\chi^m(x)))\).

Taking the Grothendieck ring of semialgebraic sets over \(K_0\), the construction of motivic volume on the ring taken over \(\bar{K}\) is refined from a geometric perspective, so to reflect the action of \(\mathrm{Gal}(\bar{K}/K_0)\). Then is obtained an explicit formula for a motivic volume of a regular scheme over \(R_0\) having special fiber with strict normal crossings. This permits to compare the motivic volume with the motivic nearby fiber of Denef and Loeser. Also, it is shown how to realize the motivic nearby fiber and the motivic Milnor fiber as motivic volumes of semialgebraic sets. In particular, it follows that they are well defined without inverting \(\mathbb{L}\).

The main result, a motivic Fubini theorem for the tropicalization map permits to compute the motivic volume for a large class of examples. It claims for an algebraic variety \(Y\) over \(K_0\), with \(\pi : \mathbb{G}^n_{m,K_0}\times_{K_0} Y \to \mathbb{G}^n_{m,K_0}\) being the projection, and for an semialgebraic set \(S\subset \mathbb{G}^n_{m,K_0}\times_{K_0} Y\) that we have \((\mathrm{trop} \circ\pi)_* 1_S\) to be a constructible function. Moreover, the motivic volume of \(S\) is equal to its motivic integral, belonging to \(\mathbb{K}^{\hat{\mu}}(Var_k)\).

From it are obtained new proofs of the two conjectures, in fact, with stronger results valid in \(\mathbb{K}^{\hat{\mu}}(Var_k)\) without inverting \(\mathbb{L}\). In particular, the proof of Kontsevich-Soibelman conjecture, first proposed in [L. Q. Thuong, Duke Math. J. 164, No. 1, 157–194 (2015; Zbl 1370.14017)] is essentially simplified and generalized. This is based by an interpretation of the motivic nearby fiber as the motivic volume of the semialgebraic nearby fiber.

Finally, all constructions are refined in the relative case. It is shown also how to generalize the conjectures to the case of invariant subvarieties of toric varieties with a \(\mathbb{G}_m\) action.

Supposed the field \(k\) is of characteristic \(0\), containing all roots of unity. Let \(R = k[[t]]\), \(K_0 = k((t))\subset K\) be the field of Puiseux series, and pick up \(\bar{K}\) an algebraic closure. We have \(\hat{\mu}\simeq \mathrm{Gal} (K/K_0)\), and let \(\bar{R}\subset\bar{K}\) be the valuation rings with respect to the \(t\)-adic valuation. For \(\mathbb{T}\) a split \(R\)-torus with character lattice \(M\) of finite rank, define the tropicalization map \(\mathrm{trop} :\mathbb{T}(\bar{K})\to \mathrm{Hom}(M, \mathbb{Q})\), \(x\mapsto (m \mapsto \mathrm{val}(\chi^m(x)))\).

Taking the Grothendieck ring of semialgebraic sets over \(K_0\), the construction of motivic volume on the ring taken over \(\bar{K}\) is refined from a geometric perspective, so to reflect the action of \(\mathrm{Gal}(\bar{K}/K_0)\). Then is obtained an explicit formula for a motivic volume of a regular scheme over \(R_0\) having special fiber with strict normal crossings. This permits to compare the motivic volume with the motivic nearby fiber of Denef and Loeser. Also, it is shown how to realize the motivic nearby fiber and the motivic Milnor fiber as motivic volumes of semialgebraic sets. In particular, it follows that they are well defined without inverting \(\mathbb{L}\).

The main result, a motivic Fubini theorem for the tropicalization map permits to compute the motivic volume for a large class of examples. It claims for an algebraic variety \(Y\) over \(K_0\), with \(\pi : \mathbb{G}^n_{m,K_0}\times_{K_0} Y \to \mathbb{G}^n_{m,K_0}\) being the projection, and for an semialgebraic set \(S\subset \mathbb{G}^n_{m,K_0}\times_{K_0} Y\) that we have \((\mathrm{trop} \circ\pi)_* 1_S\) to be a constructible function. Moreover, the motivic volume of \(S\) is equal to its motivic integral, belonging to \(\mathbb{K}^{\hat{\mu}}(Var_k)\).

From it are obtained new proofs of the two conjectures, in fact, with stronger results valid in \(\mathbb{K}^{\hat{\mu}}(Var_k)\) without inverting \(\mathbb{L}\). In particular, the proof of Kontsevich-Soibelman conjecture, first proposed in [L. Q. Thuong, Duke Math. J. 164, No. 1, 157–194 (2015; Zbl 1370.14017)] is essentially simplified and generalized. This is based by an interpretation of the motivic nearby fiber as the motivic volume of the semialgebraic nearby fiber.

Finally, all constructions are refined in the relative case. It is shown also how to generalize the conjectures to the case of invariant subvarieties of toric varieties with a \(\mathbb{G}_m\) action.

Reviewer: Peter Petrov (Maceio)

### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14E18 | Arcs and motivic integration |

14T20 | Geometric aspects of tropical varieties |

14T90 | Applications of tropical geometry |

### Keywords:

motivic integration; nearby cycles; Donaldson-Thomas Theory; tropical geometry; semialgebraic set; motivic nearby fiber; motivic volume; tropicalization map
PDF
BibTeX
XML
Cite

\textit{J. Nicaise} and \textit{S. Payne}, Duke Math. J. 168, No. 10, 1843--1886 (2019; Zbl 1429.14033)

### References:

[1] | K. Behrend, J. Bryan, and B. Szendrői, Motivic degree zero Donaldson-Thomas invariants, Invent. Math. 192 (2013), no. 1, 111-160. · Zbl 1267.14008 |

[2] | F. Bittner, On motivic zeta functions and the motivic nearby fiber, Math. Z. 249 (2005), no. 1, 63-83. · Zbl 1085.14020 |

[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. · Zbl 1415.14006 |

[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. · Zbl 1430.14105 |

[6] | B. Davison and S. Meinhardt, The motivic Donaldson-Thomas invariants of \((-2)\)-curves, Algebra Number Theory 11 (2017), no. 6, 1243-1286. · Zbl 1409.14092 |

[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. · Zbl 1079.14003 |

[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. · Zbl 1318.14061 |

[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. · Zbl 1173.14301 |

[10] | E. Hrushovski and D. Kazhdan, “Integration in valued fields” in Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser, Boston, 2006, 261-405. · Zbl 1136.03025 |

[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. · Zbl 1400.14015 |

[12] | M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435v1 [math.AG]. · Zbl 1248.14060 |

[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. · Zbl 1230.14014 |

[15] | J. Nicaise, Geometric criteria for tame ramification, Math. Z. 273 (2013), nos. 3-4, 839-868. · Zbl 1329.11067 |

[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. · Zbl 1430.14037 |

[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. · Zbl 1272.14010 |

[19] | J. Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), no. 4, 1087-1104. · Zbl 1154.14039 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.