On the invariance of Welschinger invariants

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.