Symplectic instability of Bézout’s theorem

We investigate the failure of Bézout’s Theorem for two symplectic surfaces in ℂP2 (and more generally on an algebraic surface), by proving that every plane algebraic curve C can be perturbed in the -topology to an arbitrarily close smooth symplectic surface Cϵ with the property that the cardinality #Cϵ ∩ Zd of the transversal intersection of Cϵ with an algebraic plane curve Zd of degree d, as a function of d, can grow arbitrarily fast. As a consequence we obtain that, although Bézout’s Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is “arbitrarly false” for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon “instability of Bézout’s Theorem”).