×

New cases of logarithmic equivalence of Welschinger and Gromov-Witten invariants. (English) Zbl 1203.14065

Proc. Steklov Inst. Math. 258, 65-73 (2007) and Tr. Mat. Inst. Steklova 258, 70-78 (2007).
Welschinger invariants and Gromov-Witten invariants of toric unnodal Del Pezzo surfaces can be computed with methods from tropical geometry [G. Mikhalkin, J. Am. Math. Soc. 18, No. 2, 313–377 (2005; Zbl 1092.14068); E. Shustin, J. Algebr. Geom. 15, No. 2, 285–322 (2006; Zbl 1118.14059)]. This fact is used by I. V. Itenberg, V. M. Kharlamov and E. I. Shustin [Russ. Math. Surv. 59, No. 6, 1093–1116 (2004; Zbl 1086.14047)] to prove their equivalence in the logarithmic scale for toric unnodal Del Pezzo surfaces equipped with the tautological real structure. This paper is concerned with four toric unnodal Del Pezzo surfaces with a non-tautological real structure.
In [E. Shustin, Proc. Steklov Inst. Math. 258, 218–246 (2007; Zbl 1184.14083)] it was shown how the Welschinger invariants of those can be computed with tropical methods. This result is used here to prove the logarithmic equivalence of Welschinger invariants and Gromov-Witten invariants also for the four surfaces in question. To deduce this result, the authors relate the tropical count to certain combinatorial objects called marked admissible proper systems following ideas of [Zbl 1184.14083]. Then they construct enough such proper systems in each case to deduce the logarithmic equivalence to the Gromov-Witten invariants.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
14P99 Real algebraic and real-analytic geometry

References:

[1] I. Itenberg, V. Kharlamov, and E. Shustin, ”Welschinger Invariant and Enumeration of Real Rational Curves,” Int. Math. Res. Not. 2003(49), 2639–2653 (2003). · Zbl 1083.14523 · doi:10.1155/S1073792803131352
[2] I. V. Itenberg, V. M. Kharlamov, and E. I. Shustin, ”Logarithmic Equivalence of Welschinger and Gromov-Witten Invariants,” Usp. Mat. Nauk 59(6), 85–110 (2004) [Russ. Math. Surv. 59, 1093–1116 (2004)]. · Zbl 1086.14047 · doi:10.4213/rm797
[3] I. Itenberg, V. Kharlamov, and E. Shustin, ”Logarithmic Asymptotics of the Genus Zero Gromov-Witten Invariants of the Blown up Plane,” Geom. Topol. 9, 483–491 (2005). · Zbl 1078.53090 · doi:10.2140/gt.2005.9.483
[4] G. Mikhalkin, ”Counting Curves via Lattice Paths in Polygons,” C. R., Math., Acad. Sci. Paris 336(8), 629–634 (2003). · Zbl 1027.14026 · doi:10.1016/S1631-073X(03)00104-3
[5] G. Mikhalkin, ”Enumerative Tropical Algebraic Geometry in \(\mathbb{R}\)2,” J. Am. Math. Soc. 18(2), 313–377 (2005). · Zbl 1092.14068 · doi:10.1090/S0894-0347-05-00477-7
[6] E. Shustin, ”A Tropical Approach to Enumerative Geometry,” Algebra Anal. 17(2), 170–214 (2005) [St. Petersburg Math. J. 17, 343–375 (2006)].
[7] E. Shustin, ”A Tropical Calculation of the Welschinger Invariants of Real Toric Del Pezzo Surfaces,” J. Algebr. Geom. 15(2), 285–322 (2006). · Zbl 1118.14059 · doi:10.1090/S1056-3911-06-00434-6
[8] E. Shustin, ”Welslchinger Invariants of Toric Del Pezzo Surfaces with Nonstandard Real Structures,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 258, 227–255 (2007) [Proc. Steklov Inst. Math. 258, 218–246 (2007)].
[9] J.-Y. Welschinger, ”Invariants of Real Rational Symplectic 4-Manifolds and Lower Bounds in Real Enumerative Geometry,” C. R., Math., Acad. Sci. Paris 336(4), 341–344 (2003). · Zbl 1042.57018 · doi:10.1016/S1631-073X(03)00059-1
[10] J.-Y. Welschinger, ”Invariants of Real Symplectic 4-Manifolds and Lower Bounds in Real Enumerative Geometry,” Invent. Math. 162(1), 195–234 (2005). · Zbl 1082.14052 · doi:10.1007/s00222-005-0445-0
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.