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Behavior of Welschinger invariants under Morse simplifications

We relate Welschinger invariants of a rational real symplectic 4-manifold before and after a Morse simplification (i.e deletion of a sphere or a handle of the real part of the surface). This relation is a consequence of a real version of Abramovich-Bertram formula which computes Gromov-Witten invariants by means of enumeration of J-holomorphic curves with a non-generic almost complex structure J. In addition, we give some qualitative consequences of our study, for example the vanishing of Welschinger invariants in some cases.