Geometry & Topology

Tropical refined curve counting via motivic integration

Abstract

We propose a geometric interpretation of Block and Göttsche’s refined tropical curve counting invariants in terms of virtual $χ − y$ specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from $χ − y$–genus to Euler characteristic.

Article information

Source
Geom. Topol., Volume 22, Number 6 (2018), 3175-3234.

Dates
Revised: 6 February 2018
Accepted: 27 March 2018
First available in Project Euclid: 29 September 2018

https://projecteuclid.org/euclid.gt/1538186736

Digital Object Identifier
doi:10.2140/gt.2018.22.3175

Mathematical Reviews number (MathSciNet)
MR3858763

Zentralblatt MATH identifier
06945125

Citation

Nicaise, Johannes; Payne, Sam; Schroeter, Franziska. Tropical refined curve counting via motivic integration. Geom. Topol. 22 (2018), no. 6, 3175--3234. doi:10.2140/gt.2018.22.3175. https://projecteuclid.org/euclid.gt/1538186736

References

• D Abramovich, L Caporaso, S Payne, The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. 48 (2015) 765–809
• M Baker, S Payne, J Rabinoff, Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom. 3 (2016) 63–105
• A Beauville, Counting rational curves on K$3$ surfaces, Duke Math. J. 97 (1999) 99–108
• V G Berkovich, Étale cohomology for non–Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. 78 (1993) 5–161
• V G Berkovich, Complex analytic vanishing cycles for formal schemes, preprint (2015) Available at \setbox0\makeatletter\@url http://www.wisdom.weizmann.ac.il/~ vova/FormIV_2015.pdf {\unhbox0
• V G Berkovich, Finiteness theorems for vanishing cycles of formal schemes, Israel J. Math. 210 (2015) 147–191
• F Block, L Göttsche, Refined curve counting with tropical geometry, Compos. Math. 152 (2016) 115–151
• L A Borisov, The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom. 27 (2018) 203–209
• J Denef, F Loeser, Geometry on arc spaces of algebraic varieties, from “European Congress of Mathematics, I” (C Casacuberta, R M Miró-Roig, J Verdera, S Xambó-Descamps, editors), Progr. Math. 201, Birkhäuser, Basel (2001) 327–348
• L van den Dries, o-minimal structures and real analytic geometry, from “Current developments in mathematics, 1998” (B Mazur, W Schmid, S T Yau, D Jerison, I Singer, D Stroock, editors), International, Somerville, MA (1999) 105–152
• B Fantechi, L Göttsche, D van Straten, Euler number of the compactified Jacobian and multiplicity of rational curves, J. Algebraic Geom. 8 (1999) 115–133
• S A Filippini, J Stoppa, Block–Göttsche invariants from wall-crossing, Compos. Math. 151 (2015) 1543–1567
• A Gathmann, H Markwig, Kontsevich's formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008) 537–560
• R Gopakumar, C Vafa, M–theory and topological strings, I, preprint (1998)
• R Gopakumar, C Vafa, M–theory and topological strings, II, preprint (1998)
• L Göttsche, V Shende, Refined curve counting on complex surfaces, Geom. Topol. 18 (2014) 2245–2307
• M Gross, Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics 114, Amer. Math. Soc., Providence, RI (2011)
• A Grothendieck, Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967) 5–361
• W Gubler, A guide to tropicalizations, from “Algebraic and combinatorial aspects of tropical geometry” (E Brugallé, M A Cueto, A Dickenstein, E-M Feichtner, I Itenberg, editors), Contemp. Math. 589, Amer. Math. Soc., Providence, RI (2013) 125–189
• J Harris, On the Severi problem, Invent. Math. 84 (1986) 445–461
• E Hrushovski, D Kazhdan, Integration in valued fields, from “Algebraic geometry and number theory” (V Ginzburg, editor), Progr. Math. 253, Birkhäuser, Boston (2006) 261–405
• E Hrushovski, F Loeser, Monodromy and the Lefschetz fixed point formula, Ann. Sci. Éc. Norm. Supér. 48 (2015) 313–349
• L Illusie, Théorie de Brauer et caractéristique d'Euler–Poincaré (d'après P Deligne), from “The Euler–Poincaré characteristic”, Astérisque 82, Soc. Math. France, Paris (1981) 161–172
• L Illusie, K Kato, C Nakayama, Quasi-unipotent logarithmic Riemann–Hilbert correspondences, J. Math. Sci. Univ. Tokyo 12 (2005) 1–66
• I Itenberg, V Kharlamov, E Shustin, A Caporaso–Harris type formula for Welschinger invariants of real toric del Pezzo surfaces, Comment. Math. Helv. 84 (2009) 87–126
• M Kapranov, The elliptic curve in the S–duality theory and Eisenstein series for Kac–Moody groups, preprint (2000)
• E Katz, Tropical realization spaces for polyhedral complexes, from “Algebraic and combinatorial aspects of tropical geometry” (E Brugallé, M A Cueto, A Dickenstein, E-M Feichtner, I Itenberg, editors), Contemp. Math. 589, Amer. Math. Soc., Providence, RI (2013) 235–251
• E Katz, S Payne, Realization spaces for tropical fans, from “Combinatorial aspects of commutative algebra and algebraic geometry” (G Fl\oystad, T Johnsen, A L Knutsen, editors), Abel Symp. 6, Springer (2011) 73–88
• E Katz, A Stapledon, Tropical geometry and the motivic nearby fiber, Compos. Math. 148 (2012) 269–294
• E Katz, A Stapledon, Tropical geometry, the motivic nearby fiber, and limit mixed Hodge numbers of hypersurfaces, Res. Math. Sci. 3 (2016) art id. 10
• M Kool, V Shende, R P Thomas, A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011) 397–406
• M Luxton, Z Qu, Some results on tropical compactifications, Trans. Amer. Math. Soc. 363 (2011) 4853–4876
• F Martin, Cohomology of locally closed semi-algebraic subsets, Manuscripta Math. 144 (2014) 373–400
• G Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb R^2$, J. Amer. Math. Soc. 18 (2005) 313–377
• G Mikhalkin, Quantum indices and refined enumeration of real plane curves, Acta Math. 219 (2017) 135–180
• J Nicaise, Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology, J. Algebraic Geom. 20 (2011) 199–237
• J Nicaise, J Sebag, Motivic Serre invariants, ramification, and the analytic Milnor fiber, Invent. Math. 168 (2007) 133–173
• J Nicaise, J Sebag, The Grothendieck ring of varieties, from “Motivic integration and its interactions with model theory and non-Archimedean geometry, I” (R Cluckers, J Nicaise, J Sebag, editors), London Math. Soc. Lecture Note Ser. 383, Cambridge Univ. Press (2011) 145–188
• T Nishinou, Correspondence theorems for tropical curves, I, preprint (2009)
• T Nishinou, B Siebert, Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006) 1–51
• B Osserman, S Payne, Lifting tropical intersections, Doc. Math. 18 (2013) 121–175
• R Pandharipande, R P Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010) 267–297
• A Robinson, Complete theories, North-Holland, Amsterdam (1956)
• J-P Serre, Représentations linéaires des groupes finis, Hermann, Paris (1967)
• A J Stewart, V Vologodsky, Motivic integral of K$3$ surfaces over a non-archimedean field, Adv. Math. 228 (2011) 2688–2730
• I Tyomkin, Tropical geometry and correspondence theorems via toric stacks, Math. Ann. 353 (2012) 945–995
• S-T Yau, E Zaslow, BPS states, string duality, and nodal curves on K$3$, Nuclear Phys. B 471 (1996) 503–512