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”).

中文翻译：

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Bézout 定理的辛不稳定性

摘要

我们通过证明每个平面代数曲线C^都可以在\({{\cal C}^\infty }\ ) -任意闭合光滑辛曲面C _{ϵ 的}拓扑，其属性为C _ϵ与度为d的代数平面曲线Z _d的横向交集的基数 # C _ϵ ∩ Z _d作为d的函数可以增长任意快。因此，我们得到，虽然贝祖特定理对于关于相同的几乎复杂结构的伪全纯曲线是正确的，但是对于关于不同（但任意接近）的几乎复杂结构的伪全纯曲线来说，它是“任意错误的”（我们称之为现象“贝祖定理的不稳定性”）。