Symplectic Instability of B\'ezout's Theorem

Abstract

We investigate the failure of B\'ezout's Theorem for two symplectic surfaces in $\mathbb{C}\mathrm{P}^2$ (and more generally on an algebraic surface), by proving that every plane algebraic curve $C$ can be perturbed in the $\mathscr{C}^1$-topology to an arbitrarily close smooth symplectic surface $C_\epsilon$ with the property that the cardinality $\#C_\epsilon \cap Z_d$ of the transversal intersection of $C_\epsilon$ 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 B\'ezout's Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is "arbitrarily false" for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon "instability of B\'ezout's Theorem").