Symplectic instability of Bézout’s theorem

Abstract

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 ${{\cal C}^\infty }$-topology to an arbitrarily close smooth symplectic surface C_ϵ with the property that the cardinality #C_ϵ ∩ Z_d of the transversal intersection of C_ϵ with an algebraic plane curve Z_d 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”).