Refined count of oriented real rational curves
Author:
Thomas Blomme
Journal:
J. Algebraic Geom. 33 (2024), 143-197
DOI:
https://doi.org/10.1090/jag/801
Published electronically:
May 5, 2023
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We introduce a quantum index for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen $2$-form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the 2017 results of Mikhalkin to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.
References
- Florian Block and Lothar Göttsche, Refined curve counting with tropical geometry, Compos. Math. 152 (2016), no. 1, 115–151. MR 3453390, DOI 10.1112/S0010437X1500754X
- Thomas Blomme, Computation of refined enumerative invariants in real and tropical geometry, PhD Thesis, 2020.
- Thomas Blomme, Computation of refined toric invariants ii, arXiv:2007.02275, 2020.
- Thomas Blomme, Refined count for rational tropical curves in arbitrary dimension, arXiv:2010.05777, 2020.
- Thomas Blomme, A tropical computation of refined toric invariants, arXiv:2001.09305, 2020.
- 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
- Pierrick Bousseau, The quantum tropical vertex, Geom. Topol. 24 (2020), no. 3, 1297–1379. MR 4157555, DOI 10.2140/gt.2020.24.1297
- S. A. Filippini and J. Stoppa, Block-Göttsche invariants from wall-crossing, Compos. Math. 151 (2015), no. 8, 1543–1567. MR 3383167, DOI 10.1112/S0010437X14007994
- 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
- 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
- Travis Mandel, Scattering diagrams, theta functions, and refined tropical curve counts, arXiv:1503.06183, 2015.
- Travis Mandel and Helge Ruddat, Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves, arXiv:1902.07183, 2019.
- Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\Bbb R^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI 10.1090/S0894-0347-05-00477-7
- Grigory Mikhalkin, Quantum indices and refined enumeration of real plane curves, Acta Math. 219 (2017), no. 1, 135–180. MR 3765660, DOI 10.4310/ACTA.2017.v219.n1.a5
- 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
- Ilya Tyomkin, Enumeration of rational curves with cross-ratio constraints, Adv. Math. 305 (2017), 1356–1383. MR 3570161, DOI 10.1016/j.aim.2016.10.010
- Jean-Yves Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234. MR 2198329, DOI 10.1007/s00222-005-0445-0
References
- Florian Block and Lothar Göttsche, Refined curve counting with tropical geometry, Compos. Math. 152 (2016), no. 1, 115–151. MR 3453390, DOI 10.1112/S0010437X1500754X
- Thomas Blomme, Computation of refined enumerative invariants in real and tropical geometry, PhD Thesis, 2020.
- Thomas Blomme, Computation of refined toric invariants ii, arXiv:2007.02275, 2020.
- Thomas Blomme, Refined count for rational tropical curves in arbitrary dimension, arXiv:2010.05777, 2020.
- Thomas Blomme, A tropical computation of refined toric invariants, arXiv:2001.09305, 2020.
- 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
- Pierrick Bousseau, The quantum tropical vertex, Geom. Topol. 24 (2020), no. 3, 1297–1379. MR 4157555, DOI 10.2140/gt.2020.24.1297
- S. A. Filippini and J. Stoppa, Block-Göttsche invariants from wall-crossing, Compos. Math. 151 (2015), no. 8, 1543–1567. MR 3383167, DOI 10.1112/S0010437X14007994
- 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
- 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
- Travis Mandel, Scattering diagrams, theta functions, and refined tropical curve counts, arXiv:1503.06183, 2015.
- Travis Mandel and Helge Ruddat, Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves, arXiv:1902.07183, 2019.
- Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb {R}^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI 10.1090/S0894-0347-05-00477-7
- Grigory Mikhalkin, Quantum indices and refined enumeration of real plane curves, Acta Math. 219 (2017), no. 1, 135–180. MR 3765660, DOI 10.4310/ACTA.2017.v219.n1.a5
- 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
- Ilya Tyomkin, Enumeration of rational curves with cross-ratio constraints, Adv. Math. 305 (2017), 1356–1383. MR 3570161, DOI 10.1016/j.aim.2016.10.010
- Jean-Yves Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234. MR 2198329, DOI 10.1007/s00222-005-0445-0
Additional Information
Thomas Blomme
Affiliation:
Université de Genève, 5-7 rue du Conseil Général, 1205 Genève, Switzerland
MR Author ID:
1497057
ORCID:
0000-0001-5881-4750
Email:
thomas.blomme@unige.ch
Received by editor(s):
July 15, 2021
Received by editor(s) in revised form:
November 25, 2021, and December 7, 2021
Published electronically:
May 5, 2023
Article copyright:
© Copyright 2023
University Press, Inc.
