On the invariance of Welschinger invariants
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- by E. Brugallé
- St. Petersburg Math. J. 32 (2021), 199-214
- DOI: https://doi.org/10.1090/spmj/1644
- Published electronically: March 2, 2021
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Abstract:
Some observations about original Welschinger invariants defined in the paper Invariants of real symplectic $4$-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234, 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.
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, and by Itenberg, Kharlamov, and Shustin 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 Refined curve counting with tropical geometry, Compos. Math. 152 (2016), no. 1, 115–151, and Refined broccoli invariants, J. Algebraic Geom. 28 (2019), no. 1, 1–41, is also discussed.
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Bibliographic Information
- E. Brugallé
- Affiliation: Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, F-44322 Nantes Cedex 3, France
- Email: erwan.brugalle@math.cnrs.fr
- Received by editor(s): December 4, 2018
- Published electronically: March 2, 2021
- Additional Notes: This work was partially supported by the grant TROPICOUNT of Région Pays de la Loire, and the ANR project ENUMGEOM NR-18-CE40-0009-02.
- © Copyright 2021 American Mathematical Society
- Journal: St. Petersburg Math. J. 32 (2021), 199-214
- MSC (2020): Primary 14P05, 14N10; Secondary 14N35, 14P25
- DOI: https://doi.org/10.1090/spmj/1644
- MathSciNet review: 4074999