Symplectic instability of Bézout’s theorem

Pub Date : 2023-12-18 DOI:10.1007/s11856-023-2598-1

引用次数: 0

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”).

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贝祖特定理的交映不稳定性

摘要我们研究了贝祖特定理在ℂP2（更广义地说，在代数曲面上）两个交映曲面上的失效、通过证明每一条平面代数曲线 C 都可以在 \({{\cal C}^\infty }\) -topology 中被扰动到一个任意接近的光滑交错曲面 Cϵ，其性质是 Cϵ 与阶数为 d 的代数平面曲线 Zd 的横交的心数 #Cϵ ∩ Zd 作为 d 的函数可以任意快速增长。因此，我们得到，虽然贝祖特定理对于同一近似复结构的伪全形曲线是真的，但对于不同（但任意接近）近似复结构的伪全形曲线却是 "任意假的"（我们称这种现象为 "贝祖特定理的不稳定性"）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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