Log BPS numbers of log Calabi-Yau surfaces

Choi, Jinwon; van Garrel, Michel; Katz, Sheldon; Takahashi, Nobuyoshi

doi:10.1090/tran/8234

Log BPS numbers of log Calabi-Yau surfaces
HTML articles powered by AMS MathViewer

by Jinwon Choi, Michel van Garrel, Sheldon Katz and Nobuyoshi Takahashi PDF

Trans. Amer. Math. Soc. 374 (2021), 687-732 Request permission

Abstract:

Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb {A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

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
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
M. Alim, M. Hecht, H. Jockers, P. Mayr, A. Mertens, and M. Soroush, Hints for off-shell mirror symmetry in type II/F-theory compactifications, Nuclear Phys. B 841 (2010), no. 3, 303–338. MR 2720856, DOI 10.1016/j.nuclphysb.2010.06.017
Denis Auroux, Mirror symmetry and $T$-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51–91. MR 2386535
L. Barrott and N. Nabijou, Tangent curves to degenerating hypersurfaces, in preparation, 2020.
Arnaud Beauville, Counting rational curves on $K3$ surfaces, Duke Math. J. 97 (1999), no. 1, 99–108. MR 1682284, DOI 10.1215/S0012-7094-99-09704-1
K. Behrend, Algebraic Gromov-Witten invariants, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 19–70. MR 1714820, DOI 10.1017/CBO9780511721540.004
P. Bousseau, A proof of N. Takahashi’s conjecture on genus zero Gromov-Witten theory of $(\mathbb {P}^2,{E})$, arXiv:1909.02992.
Pierrick Bousseau, Quantum mirrors of log Calabi-Yau surfaces and higher-genus curve counting, Compos. Math. 156 (2020), no. 2, 360–411. MR 4048291, DOI 10.1112/s0010437x19007760
P. Bousseau, The quantum tropical vertex, arXiv:1806.11495.
P. Bousseau, Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb {P}^2$, arXiv:1909.02985.
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
P. Bousseau, A. Brini and M. van Garrel, On the log-local principle for the toric boundary, arXiv:1908.04371.
P. Bousseau, A. Brini and M. van Garrel, Stable maps to Looijenga pairs, in preparation, 2020.
Jim Bryan and Rahul Pandharipande, BPS states of curves in Calabi-Yau 3-folds, Geom. Topol. 5 (2001), 287–318. MR 1825668, DOI 10.2140/gt.2001.5.287
M. Carl, M. Pumperla and B. Siebert, A tropical view on Landau-Ginzburg models, preprint available at https://www.math.uni-hamburg.de/home/siebert/research/research.html.
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
Xi Chen, A simple proof that rational curves on $K3$ are nodal, Math. Ann. 324 (2002), no. 1, 71–104. MR 1931759, DOI 10.1007/s00208-002-0329-1
J. Choi, M. van Garrel, S. Katz and N. Takahashi, Local BPS numbers: Enumerative aspects and wall-crossing, Int. Math. Res. Notices, published online on 02 August 2018, rny171, https://doi.org/10.1093/imrn/rny171.
J. Choi, M. van Garrel, S. Katz and N. Takahashi, Sheaves of maximal intersection and multiplicities of stable log maps arXiv:1908.10906.
T. Collins, A. Jacob, Y.-S. Lin, Special Lagrangian submanifolds of log Calabi-Yau manifolds, arXiv:1904.08363
Ulrich Derenthal, Michael Joyce, and Zachariah Teitler, The nef cone volume of generalized del Pezzo surfaces, Algebra Number Theory 2 (2008), no. 2, 157–182. MR 2377367, DOI 10.2140/ant.2008.2.157
L. Diogo and S. Lisi, Symplectic Homology for complements of smooth divisors, arXiv:1804.08014.
Sandra Di Rocco, $k$-very ample line bundles on del Pezzo surfaces, Math. Nachr. 179 (1996), 47–56. MR 1389449, DOI 10.1002/mana.19961790104
Honglu Fan, Hsian-Hua Tseng, and Fenglong You, Mirror theorems for root stacks and relative pairs, Selecta Math. (N.S.) 25 (2019), no. 4, Paper No. 54, 25. MR 3997137, DOI 10.1007/s00029-019-0501-z
B. Fantechi, L. Göttsche, and D. van Straten, Euler number of the compactified Jacobian and multiplicity of rational curves, J. Algebraic Geom. 8 (1999), no. 1, 115–133. MR 1658220
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Floer theory on compact toric manifolds. I, Duke Math. J. 151 (2010), no. 1, 23–174. MR 2573826, DOI 10.1215/00127094-2009-062
Michel van Garrel, Local and relative BPS state counts for del Pezzo surfaces, String-Math 2014, Proc. Sympos. Pure Math., vol. 93, Amer. Math. Soc., Providence, RI, 2016, pp. 215–220. MR 3525992
M. van Garrel, On a conjecture by N. Takahashi on log mirror symmetry for the projective plane, AMS Adv. Stud. Pure Math. no. 83, 2019, Primitive forms and related subjects - Kavli IPMU 2014, pp. 117–126.
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, D. Peter Overholser, and Helge Ruddat, Enumerative aspects of the Gross-Siebert program, Calabi-Yau varieties: arithmetic, geometry and physics, Fields Inst. Monogr., vol. 34, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, pp. 337–420. MR 3409781, DOI 10.1007/978-1-4939-2830-9_{1}1
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, Absolute and relative Gromov-Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002), no. 2, 171–203. MR 1944571, DOI 10.1215/S0012-7094-02-11521-X
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-I, arXiv:hep-th/9809187, 1998.
R. Gopakumar and C. Vafa, M-theory and topological strings-II, arXiv:hep-th/9812127, 1998.
Lothar Göttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998), no. 3, 523–533. MR 1645204, DOI 10.1007/s002200050434
Gräfnitz, Tim, Tropical correspondence for the log Calabi-Yau pair $(\mathbb {P}^2,{E})$, arXiv:2005.14018, 2020.
Mark Gross, Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, vol. 114, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2011. MR 2722115, DOI 10.1090/cbms/114
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.
M. Gross, P. Hacking, S. Keel, B. Siebert: Theta functions and K3 surfaces, in preparation.
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, 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, 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
Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724. MR 552521
M. Huang, A. Klemm, and M. Poretschkin, Refined stable pair invariants for E-, M- and $[p,q]$-strings, arXiv:1308.0619.
Dominic Joyce and Yinan Song, A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), no. 1020, iv+199. MR 2951762, DOI 10.1090/S0065-9266-2011-00630-1
Sheldon Katz, Genus zero Gopakumar-Vafa invariants of contractible curves, J. Differential Geom. 79 (2008), no. 2, 185–195. MR 2420017
Sheldon Katz, Albrecht Klemm, and Cumrun Vafa, M-theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1445–1537. MR 1796683, DOI 10.4310/ATMP.1999.v3.n5.a6
B. Kim, H. Lho, and H. Ruddat, The degeneration formula for stable log maps, arXiv:1803.04210.
A. Klemm, D. Maulik, R. Pandharipande, and E. Scheidegger, Noether-Lefschetz theory and the Yau-Zaslow conjecture, J. Amer. Math. Soc. 23 (2010), no. 4, 1013–1040. MR 2669707, DOI 10.1090/S0894-0347-2010-00672-8
Andreas Leopold Knutsen, Exceptional curves on del Pezzo surfaces, Math. Nachr. 256 (2003), 58–81. MR 1989378, DOI 10.1002/mana.200310070
Maxim Kontsevich and Yan Soibelman, Motivic Donaldson-Thomas invariants: summary of results, Mirror symmetry and tropical geometry, Contemp. Math., vol. 527, Amer. Math. Soc., Providence, RI, 2010, pp. 55–89. MR 2681792, DOI 10.1090/conm/527/10400
Martijn Kool, Vivek Shende, and Richard P. Thomas, A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011), no. 1, 397–406. MR 2776848, DOI 10.2140/gt.2011.15.397
Martijn Kool and Richard Thomas, Reduced classes and curve counting on surfaces I: theory, Algebr. Geom. 1 (2014), no. 3, 334–383. MR 3238154, DOI 10.14231/AG-2014-017
W. Lerche and P. Mayr, On N=1 Mirror Symmetry for Open Type II Strings, arXiv:hep-th/0111113.
W. Lerche, P. Mayr, and N. Warner, Holomorphic N=1 Special Geometry of Open–Closed Type II Strings, arXiv:hep-th/0207259.
Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667
J. Li, A degeneration formula of Gromov-Witten invariants, J. Diff. Geom, vol. 60, (2002), pp. 199–293.
Yu-Shen Lin, Open Gromov-Witten invariants on elliptic K3 surfaces and wall-crossing, Comm. Math. Phys. 349 (2017), no. 1, 109–164. MR 3592747, DOI 10.1007/s00220-016-2754-0
Y.-S. Lin, Correspondence Theorem between Holomorphic Discs and Tropical Discs on K3 Surfaces, to appear in J. Diff. Geom.
Jun Li and Yu-jong Tzeng, Universal polynomials for singular curves on surfaces, Compos. Math. 150 (2014), no. 7, 1169–1182. MR 3230849, DOI 10.1112/S0010437X13007756
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
Yu. I. Manin, Generating functions in algebraic geometry and sums over trees, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 401–417. MR 1363064, DOI 10.1007/978-1-4612-4264-2_{1}4
Davesh Maulik and Rahul Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory, A celebration of algebraic geometry, Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013, pp. 469–507. MR 3114953
D. Maulik, R. Pandharipande, and R. P. Thomas, Curves on $K3$ surfaces and modular forms, J. Topol. 3 (2010), no. 4, 937–996. With an appendix by A. Pixton. MR 2746343, DOI 10.1112/jtopol/jtq030
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
N. Nabijou and D. Ranganathan, Gromov-Witten theory with maximal contacts, arXiv:1908.04706.
R. Pandharipande, Hodge integrals and degenerate contributions, Comm. Math. Phys. 208 (1999), no. 2, 489–506. MR 1729095, DOI 10.1007/s002200050766
R. Pandharipande, Three questions in Gromov-Witten theory, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 503–512. MR 1957060
R. Pandharipande and R. P. Thomas, The Katz-Klemm-Vafa conjecture for $K3$ surfaces, Forum Math. Pi 4 (2016), e4, 111. MR 3508473, DOI 10.1017/fmp.2016.2
Markus Reineke, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, Compos. Math. 147 (2011), no. 3, 943–964. MR 2801406, DOI 10.1112/S0010437X1000521X
Markus Reineke, Degenerate cohomological Hall algebra and quantized Donaldson-Thomas invariants for $m$-loop quivers, Doc. Math. 17 (2012), 1–22. MR 2889742
Vivek Shende, Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation, Compos. Math. 148 (2012), no. 2, 531–547. MR 2904196, DOI 10.1112/S0010437X11007378
Jan Stienstra, Mahler measure variations, Eisenstein series and instanton expansions, Mirror symmetry. V, AMS/IP Stud. Adv. Math., vol. 38, Amer. Math. Soc., Providence, RI, 2006, pp. 139–150. MR 2282958, DOI 10.1090/amsip/038/07
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:alg-geom/9605007.
N. Takahashi, On the multiplicity of reducible relative stable morphisms, arXiv:1711.08173.
Nobuyoshi Takahashi, Log mirror symmetry and local mirror symmetry, Comm. Math. Phys. 220 (2001), no. 2, 293–299. MR 1844627, DOI 10.1007/PL00005567
Damiano Testa, Anthony Várilly-Alvarado, and Mauricio Velasco, Cox rings of degree one del Pezzo surfaces, Algebra Number Theory 3 (2009), no. 7, 729–761. MR 2579393, DOI 10.2140/ant.2009.3.729
H.-H. Tseng and F. You, A mirror theorem for multi-root stacks and applications, arXiv:2006.08991.
Yu-Jong Tzeng, A proof of the Göttsche-Yau-Zaslow formula, J. Differential Geom. 90 (2012), no. 3, 439–472. MR 2916043
Y. Tzeng, Enumeration of singular varieties with tangency conditions, arXiv:1703.02513.
Claire Voisin, A mathematical proof of a formula of Aspinwall and Morrison, Compositio Math. 104 (1996), no. 2, 135–151. MR 1421397
Jonathan Wise, Uniqueness of minimal morphisms of logarithmic schemes, Algebr. Geom. 6 (2019), no. 1, 50–63. MR 3904798, DOI 10.14231/AG-2019-003
Shing-Tung Yau and Eric Zaslow, BPS states, string duality, and nodal curves on $K3$, Nuclear Phys. B 471 (1996), no. 3, 503–512. MR 1398633, DOI 10.1016/0550-3213(96)00176-9