Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs

Graefnitz, Tim

doi:10.1090/jag/794

Author: Tim Graefnitz
Journal: J. Algebraic Geom. 31 (2022), 687-749
DOI: https://doi.org/10.1090/jag/794
Published electronically: June 24, 2022
Full-text PDF

Abstract | References | Additional Information

Abstract: Consider a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $\geq 3$ and a smooth anticanonical divisor $D$. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of $X$ intersecting $D$ in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of $(X,D)$ from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generating function for these invariants.

References

Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math. 18 (2014), no. 3, 465–488. MR 3257836, DOI 10.4310/AJM.2014.v18.n3.a5
Mohammad Akhtar, Tom Coates, Alessio Corti, Liana Heuberger, Alexander Kasprzyk, Alessandro Oneto, Andrea Petracci, Thomas Prince, and Ketil Tveiten, Mirror symmetry and the classification of orbifold del Pezzo surfaces, Proc. Amer. Math. Soc. 144 (2016), no. 2, 513–527. MR 3430830, DOI 10.1090/proc/12876
Dan Abramovich, Qile Chen, Mark Gross, and Bernd Siebert, Decomposition of degenerate Gromov-Witten invariants, Compos. Math. 156 (2020), no. 10, 2020–2075. MR 4177284, DOI 10.1112/s0010437x20007393
D. Abramovich, Q. Chen, M. Gross, and B. Siebert, Punctured logarithmic maps, arXiv:2009.07720, 2020.
Dan Abramovich and Barbara Fantechi, Orbifold techniques in degeneration formulas, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 2, 519–579. MR 3559610
Dan Abramovich, Steffen Marcus, and Jonathan Wise, Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 4, 1611–1667 (English, with English and French summaries). MR 3329675, DOI 10.5802/aif.2892
Dan Abramovich and Jonathan Wise, Birational invariance in logarithmic Gromov-Witten theory, Compos. Math. 154 (2018), no. 3, 595–620. MR 3778185, DOI 10.1112/S0010437X17007667
H. Argüz, Mirror symmetry for the Tate curve via tropical and log corals, arXiv:1712.10260, 2017.
L. J. Barrott and N. Nabijou, Tangent curves to degenerating hypersurfaces, arXiv:2007.05016, 2020.
Pierrick Bousseau, Tropical refined curve counting from higher genera and lambda classes, Invent. Math. 215 (2019), no. 1, 1–79. MR 3904449, DOI 10.1007/s00222-018-0823-z
Pierrick Bousseau, The quantum tropical vertex, Geom. Topol. 24 (2020), no. 3, 1297–1379. MR 4157555, DOI 10.2140/gt.2020.24.1297
P. Bousseau, Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb {P}^2$, J. Algebraic Geom. 31 (2022), no.4; arXiv:1909.02985, 2019.
P. Bousseau, A proof of N. Takahashi’s conjecture on genus zero Gromov-Witten theory of $(\mathbb {P}^2,E)$, arXiv:1909.02992, 2019.
P. Bousseau, A. Brini, and M. van Garrel, Stable maps to Looijenga pairs, arXiv:2011.08830, 2020.
Jim Bryan and Rahul Pandharipande, Curves in Calabi-Yau threefolds and topological quantum field theory, Duke Math. J. 126 (2005), no. 2, 369–396. MR 2115262, DOI 10.1215/S0012-7094-04-12626-0
Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, and Alexander Kasprzyk, Mirror symmetry and Fano manifolds, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2013, pp. 285–300. MR 3469127
M. Carl, M. Pumperla, and B. Siebert, A tropical view on Landau-Ginzburg models, arXiv:2205.07753, 2022.
Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs I, Ann. of Math. (2) 180 (2014), no. 2, 455–521. MR 3224717, DOI 10.4007/annals.2014.180.2.2
Qile Chen, The degeneration formula for logarithmic expanded degenerations, J. Algebraic Geom. 23 (2014), no. 2, 341–392. MR 3166394, DOI 10.1090/S1056-3911-2013-00614-1
Jinwon Choi, Michel van Garrel, Sheldon Katz, and Nobuyoshi Takahashi, Local BPS invariants: enumerative aspects and wall-crossing, Int. Math. Res. Not. IMRN 17 (2020), 5450–5475. MR 4146344, DOI 10.1093/imrn/rny171
J. Choi, M. van Garrel, S. Katz, and N. Takahashi, Log BPS numbers of log Calabi-Yau surfaces, arXiv:1810.02377, 2018.
Jinwon Choi, Michel van Garrel, Sheldon Katz, and Nobuyoshi Takahashi, Sheaves of maximal intersection and multiplicities of stable log maps, Selecta Math. (N.S.) 27 (2021), no. 4, Paper No. 61, 51. MR 4280381, DOI 10.1007/s00029-021-00671-0
T.-M. Chiang, A. Klemm, S.-T. Yau, and E. Zaslow, Local mirror symmetry: calculations and interpretations, Adv. Theor. Math. Phys. 3 (1999), no. 3, 495–565. MR 1797015, DOI 10.4310/ATMP.1999.v3.n3.a3
Kenji Fukaya, Multivalued Morse theory, asymptotic analysis and mirror symmetry, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005, pp. 205–278. MR 2131017, DOI 10.1090/pspum/073/2131017
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Michel van Garrel, Tom Graber, and Helge Ruddat, Local Gromov-Witten invariants are log invariants, Adv. Math. 350 (2019), 860–876. MR 3948687, DOI 10.1016/j.aim.2019.04.063
Michel van Garrel, Tony W. H. Wong, and Gjergji Zaimi, Integrality of relative BPS state counts of toric del Pezzo surfaces, Commun. Number Theory Phys. 7 (2013), no. 4, 671–687. MR 3228298, DOI 10.4310/CNTP.2013.v7.n4.a3
Andreas Gathmann, Relative Gromov-Witten invariants and the mirror formula, Math. Ann. 325 (2003), no. 2, 393–412. MR 1962055, DOI 10.1007/s00208-002-0345-1
R. Gopakumar and C. Vafa, M-theory and topological strings II, arXiv:9812127, 1998.
T. Gräfnitz, Theta functions, broken lines and $2$-marked log Gromov-Witten invariants, arXiv:2204.12257, 2022.
T. Gräfnitz, H. Ruddat, and E. Zaslow, The proper Landau-Ginzburg potential is the open mirror map, arXiv:2204.12249, 2022.
Mark Gross, Mirror symmetry for $\Bbb P^2$ and tropical geometry, Adv. Math. 224 (2010), no. 1, 169–245. MR 2600995, DOI 10.1016/j.aim.2009.11.007
M. Gross, Mirror symmetry and tropical geometry, Regional Conference Series in Mathematics 144, Springer, 2011, ISBN: 978-0-8218-5232-3.
Mark Gross, Paul Hacking, and Sean Keel, Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168. MR 3415066, DOI 10.1007/s10240-015-0073-1
M. Gross, P. Hacking, and B. Siebert, Theta functions on varieties with effective anti-canonical class, arXiv:1601.07081, 2016.
Mark Gross, Rahul Pandharipande, and Bernd Siebert, The tropical vertex, Duke Math. J. 153 (2010), no. 2, 297–362. MR 2667135, DOI 10.1215/00127094-2010-025
Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), no. 2, 169–338. MR 2213573
Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010), no. 4, 679–780. MR 2669728, DOI 10.1090/S1056-3911-2010-00555-3
Mark Gross and Bernd Siebert, From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), no. 3, 1301–1428. MR 2846484, DOI 10.4007/annals.2011.174.3.1
Mark Gross and Bernd Siebert, An invitation to toric degenerations, Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, Surv. Differ. Geom., vol. 16, Int. Press, Somerville, MA, 2011, pp. 43–78. MR 2893676, DOI 10.4310/SDG.2011.v16.n1.a2
Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. MR 3011419, DOI 10.1090/S0894-0347-2012-00757-7
Mark Gross and Bernd Siebert, Intrinsic mirror symmetry and punctured Gromov-Witten invariants, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 199–230. MR 3821173
M. Gross and B. Siebert, Intrinsic mirror symmetry, arXiv:1909.07649, 2019.
Eleny-Nicoleta Ionel and Thomas H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935–1025. MR 2113018, DOI 10.4007/annals.2004.159.935
Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
Fumiharu Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), no. 2, 215–232. MR 1754621, DOI 10.1142/S0129167X0000012X
B. Kim, H. Lho, and H. Ruddat, The degeneration formula for stable log maps, arXiv:1803.04210, 2018.
Yukiko Konishi and Satoshi Minabe, Local Gromov-Witten invariants of cubic surfaces via nef toric degeneration, Ark. Mat. 47 (2009), no. 2, 345–360. MR 2529706, DOI 10.1007/s11512-007-0064-7
Maxim Kontsevich and Yan Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 321–385. MR 2181810, DOI 10.1007/0-8176-4467-9_{9}
An-Min Li and Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218. MR 1839289, DOI 10.1007/s002220100146
Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667
Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
Y.-S. Lin, Enumerative geometry of del Pezzo surfaces, arXiv:2005.08681, 2020.
Travis Mandel and Helge Ruddat, Descendant log Gromov-Witten invariants for toric varieties and tropical curves, Trans. Amer. Math. Soc. 373 (2020), no. 2, 1109–1152. MR 4068259, DOI 10.1090/tran/7936
David Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272. MR 352106
Takeo Nishinou and Bernd Siebert, Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), no. 1, 1–51. MR 2259922, DOI 10.1215/S0012-7094-06-13511-1
Logarithmic enumerative geometry and mirror symmetry, Oberwolfach Rep. 16 (2019), no. 2, 1639–1695. Abstracts from the workshop held June 16–22, 2019; Organized by Dan Abramovich, Michel van Garrel and Helge Ruddat. MR 4109324, DOI 10.4171/OWR/2019/27
Thomas Prince, Smoothing toric Fano surfaces using the Gross-Siebert algorithm, Proc. Lond. Math. Soc. (3) 117 (2018), no. 3, 617–660. MR 3857695, DOI 10.1112/plms.12153
Thomas Prince, The tropical superpotential for $\Bbb P^2$, Algebr. Geom. 7 (2020), no. 1, 30–58. MR 4038403, DOI 10.14231/ag-2020-002
M. Pumperla, Unifying constructions in toric mirror symmetry, PhD Thesis, 2011.
Matthias Schütt and Tetsuji Shioda, Elliptic surfaces, Algebraic geometry in East Asia—Seoul 2008, Adv. Stud. Pure Math., vol. 60, Math. Soc. Japan, Tokyo, 2010, pp. 51–160. MR 2732092, DOI 10.2969/aspm/06010051
Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831, DOI 10.1016/0550-3213(96)00434-8
N. Takahashi, Curves in the complement of a smooth plane cubic whose normalizations are $\mathbb {A}^1$, arXiv:9605007, 1996.
Nobuyoshi Takahashi, Log mirror symmetry and local mirror symmetry, Comm. Math. Phys. 220 (2001), no. 2, 293–299. MR 1844627, DOI 10.1007/PL00005567

Additional Information

Tim Graefnitz
Affiliation: Department of Mathematics, University of Hamburg, Germany
Email: tim.graefnitz@gmx.de

Received by editor(s): October 6, 2020
Received by editor(s) in revised form: September 8, 2021, October 15, 2021, and March 10, 2022
Published electronically: June 24, 2022
Additional Notes: This work was supported by the DFG funded Research Training Group 1670 “Mathematics Inspired by String theory and Quantum Field Theory”
Article copyright: © Copyright 2022 University Press, Inc.