Symplectic Instability of B\'ezout's Theorem

TL;DR

This paper demonstrates that Bézout's Theorem, which holds for pseudoholomorphic curves with the same almost complex structure, can fail arbitrarily for different, close structures in symplectic geometry, revealing an instability phenomenon.

Contribution

It proves that Bézout's Theorem can be arbitrarily violated for symplectic surfaces under small perturbations of the almost complex structure.

Findings

Constructed symplectic surfaces with intersection counts growing arbitrarily fast.

Showed Bézout's Theorem is stable for fixed almost complex structures.

Established the phenomenon of instability of Bézout's Theorem under perturbations.

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").