On the invariance of Welschinger invariants. (English) Zbl 1470.14116
St. Petersbg. Math. J. 32, No. 2, 199-214 (2021) and Algebra Anal. 32, No. 2, 1-20 (2020).
Summary: Some observations about original J.-Y. Welschinger invariants defined in the paper [Invent. Math. 162, No. 1, 195–234 (2005; Zbl 1082.14052)], are collected. None of their proofs is difficult, nevertheless these remarks do not seem to have been made before. The main result is that when \(X_{\mathbb{R}}\) is a real rational algebraic surface, Welschinger invariants only depend on the number of real interpolated points, and on some homological data associated with \(X_{\mathbb{R}}\). This strengthened invariance statement was initially proved by Welschinger [loc. cit.]. This main result follows easily from a formula relating Welschinger invariants of two real symplectic manifolds that differ by a surgery along a real Lagrangian sphere. In its turn, once one believes that such a formula may hold, its proof is a mild adaptation of the proof of analogous formulas previously obtained by the author on the one hand [J. Singul. 17, 267–294 (2018; Zbl 1423.14328)], and by I. Itenberg et al. [Int. J. Math. 26, No. 8, Article ID 1550060, 63 p. (2015; Zbl 1351.14035)] on the other hand. The two aforementioned results are applied to complete the computation of Welschinger invariants of real rational algebraic surfaces, and to obtain vanishing, sign, and sharpness results for these invariants, which generalize previously known statements. Some hypothetical relationship of the present work with tropical refined invariants defined in the papers [F. Block and L. Göttsche, Compos. Math. 152, No. 1, 115–151 (2016; Zbl 1348.14125)], and [L. Göttsche and F. Schroeter, J. Algebr. Geom. 28, No. 1, 1–41 (2019; Zbl 1439.14183)], is also discussed.
MSC:
| 14P05 | Real algebraic sets |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14P25 | Topology of real algebraic varieties |
| 14T20 | Geometric aspects of tropical varieties |
| 53D05 | Symplectic manifolds (general theory) |
Keywords:
real enumerative geometry; Welschinger invariants; real rational algebraic surfaces; refined invariantsReferences:
| [1] | A. Arroyo, E. Brugall\'e, and L. L\'opez de Medrano, Recursive formula for Welschinger invariants, Int. Math. Res. Not. IMRN 5 (2011), 1107-1134. · Zbl 1227.14047 |
| [2] | F. Block and L. G\"ottsche, Refined curve counting with tropical geometry, Compos. Math. 152 (2016), no. 1, 115-151. · Zbl 1348.14125 |
| [3] | F. Block, A. Gathmann, and H Markwig, Psi-floor diagrams and a Caporaso-Harris type recursion, Israel J. Math. 191 (2012), no. 1, 405-449. · Zbl 1257.14041 |
| [4] | E. Brugall\'e, I. Itenberg, G. Mikhalkin, and K. Shaw, Brief introduction to tropical geometry, Proc. of the G\"okova Geometry-Topology Conference 2014, G\"okova Geometry/Topology Conference (GGT), G\"okova, 2015, pp. 1-75. · Zbl 1354.14089 |
| [5] | E. Brugall\'e and G. Mikhalkin, Floor decompositions of tropical curves: the planar case, Proc. of \(15\) th G\"okova Geometry-Topology Conference 2008, G\"okova Geometry/Topology Conference (GGT), G\"okova, 2009, pp. 64-90. · Zbl 1200.14106 |
| [6] | E. Brugall\'e and H. Markwig, Deformation and tropical Hirzebruch surfaces and enumerative geometry, J. Algebraic Geom. 25 (2016), no. 4, 633-702. · Zbl 1396.14065 |
| [7] | P. Bousseau, Tropical refined curve counting from higher genera and lambda classes, Invent. Math. 215 (2019), no. 1, 1-79. · Zbl 07015696 |
| [8] | \bysame , Refined floor diagrams from higher genera and lambda classes, arXiv:1904.10311, 2019. |
| [9] | E. Brugall\'e and N. Puignau, Behavior of Welschinger invariants under Morse simplifications, Rend. Semin. Mat. Univ. Padova 130 (2013), 147-153. · Zbl 1292.14034 |
| [10] | \bysame , On Welschinger invariants of symplectic \(4\)-manifolds, Comment. Math. Helv. 90 (2015), no. 4, 905-938. · Zbl 1353.53090 |
| [11] | E. Brugall\'e, Floor diagrams relative to a conic, and GW-W invariants of del Pezzo surfaces, Adv. Math. 279 (2015), 438-500. · Zbl 1316.14104 |
| [12] | \bysame , Surgery of real symplectic fourfolds and welschinger invariants, J. Singularities 17 (2018), 267-294. · Zbl 1423.14328 |
| [13] | L. Blechman and E. Shustin, Refined descendant invariants of toric surfaces, Discrete Comput. Geom. 62 (2019), no. 1, 180-208. · Zbl 1423.14347 |
| [14] | X. Chen, Solomon’s relations for Welshinger’s invariants: Examples, arXiv:1809.08938, 2018. |
| [15] | Y. Ding and J. Hu, Welschinger invariants of blow-ups of symplectic \(4\)-manifolds, Rocky Mountain J. Math. 48 (2018), no. 4, 1105-1144. · Zbl 1401.53076 |
| [16] | A. I. Degtyarev and V. M. Kharlamov, Topological properties of real algebraic varieties: Rokhlin’s way, Uspekhi Mat. Nauk 55 (2000), no. 4, 129-212; English transl., Russian Math. Surveys 55 (2000), no. 4, 735-814. · Zbl 1014.14030 |
| [17] | \bysame , Real rational surfaces are quasi-simple, J. Reine Angew. Math. 551 (2002), 87-99. · Zbl 1003.14010 |
| [18] | P. Georgieva, Open Gromov-Witten disk invariants in the presence of an anti-symplectic involution, Adv. Math. 301 (2016), 116-160. · Zbl 1365.53082 |
| [19] | L. G\"ottsche and V. Shende, Refined curve counting on complex surfaces, Geom. Topol. 18 (2014), no. 4, 2245-2307. · Zbl 1310.14012 |
| [20] | L. G\"ottsche and F. Schroeter, Refined broccoli invariants, J. Algebraic Geom. 28 (2019), no. 1, 1-41. · Zbl 1439.14183 |
| [21] | P. Georgieva and A. Zinger, Real Gromov-Witten theory in all genera and real enumerative geometry: construction, Ann. of Math. (2) 188 (2018), no. 3, 685-752. · Zbl 1432.53128 |
| [22] | A. Horev A. and J. Solomon, The open Gromov-Witten-Welschinger theory of blowups of the projective plane, arXiv:1210.4034, 2012. |
| [23] | I. Itenberg, V. Kharlamov, and E. Shustin, Logarithmic equivalence of Welschinger and Gromov-Witten invariants, Uspekhi Mat. Nauk 59 (2004), no. 6, 85-110; English transl., Russian Math. Surveys 59 (2004), no. 6, 1093-1116. · Zbl 1086.14047 |
| [24] | \bysame , Welschinger invariants of real del Pezzo surfaces of degree \(\geq 3\), Math. Ann. 355 (2013), no. 3, 849-878. · Zbl 1308.14058 |
| [25] | \bysame , Welschinger invariants of real del Pezzo surfaces of degree \(\geq 2\), Internat. J. Math. 26 (2015), no. 8, 1550060. · Zbl 1351.14035 |
| [26] | \bysame , Welschinger invariants revisited, Analysis meets geometry, Trends Math., Birkh\"auser/Springer, Cham, 2017, pp. 239-260. · Zbl 1402.14074 |
| [27] | I. Itenberg and G. Mikhalkin, On Block-G\"ottsche multiplicities for planar tropical curves, Int. Math. Res. Not. IMRN 23 (2013), 5289-5320. · Zbl 1329.14114 |
| [28] | E.-N. Ionel and T. H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935-1025. · Zbl 1075.53092 |
| [29] | J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199-293. · Zbl 1063.14069 |
| [30] | \bysame , Lecture notes on relative GW-invariants, Intersection theory and moduli, ICTP Lect. Notes, vol. XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 41-96. · Zbl 1092.14067 |
| [31] | A. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau \(3\)\nobreakdash -folds, Invent. Math. 145 (2001), no. 1, 151-218. · Zbl 1062.53073 |
| [32] | G. Mikhalkin, Enumerative tropical algebraic geometry in \(\mathbb R^2\), J. Amer. Math. Soc. 18 (2005), no. 2, 313-377. · Zbl 1092.14068 |
| [33] | \bysame , Quantum indices and refined enumeration of real plane curves, Acta Math. 219 (2017), no. 1, 135-180. · Zbl 1468.14092 |
| [34] | J. Nicaise, S. Payne, and F. Schroeter, Tropical refined curve counting via motivic integration, Geom. Topol. 22 (2018), no. 6, 3175-3234. · Zbl 1430.14037 |
| [35] | E. Shustin, On higher genus Welschinger invariants of del Pezzo surfaces, Intern. Math. Res. Not. IMRN 16 (2015), 6907-6940. · Zbl 1349.14130 |
| [36] | M. F. Tehrani and A. Zinger, On symplectic sum formulas in Gromov-Witten theory, arXiv:1404.1898, 2014. |
| [37] | J. Y. Welschinger, Invariants of real symplectic \(4\)-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195-234. · Zbl 1082.14052 |
| [38] | \bysame , Optimalit\'e, congruences et calculs d’invariants des vari\'et\'es symplectiques r\'eelles de dimension quatre, arXiv:0707.4317, 2007. |
| [39] | \bysame , Open Gromov-Witten invariants in dimension four, J. Symplectic Geom. 13 (2015), no. 4, 1075-1100. · Zbl 1339.53088 |
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.
