Dynamic Intersections

Abstract

Let X S Y be a regular imbedding of codimension d, with normal bundle N _X Y; let V be a k-dimensional subvariety of Y, W= X∩V,N the restriction of N _X Y to W, and C⊂N the normal cone to W in V. In Chap. 6 the intersection class X·V in A _k-d (W) has been constructed to be $s\frac{ * }{N}\left[ C \right]$ where s _N W → Nis the zero-section.

If X S Y is imbedded in a family S S Y x T of regular imbeddings, with T a non-singular curve, 0 ∈ T, X₀ = X, and S ⊂ Y x T is a deformation of V, then there is a closed set $\left( {\begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}{X_t} \cap {V_t}} \right)$ contained in W, and a class we denote $\begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right)$ in ${A_{k - d}}\left( {\begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}{X_t} \cap {V_t}} \right)$ which refines X·V, i.e., maps to X·V in A _k-d (W)

The Kodaira-Spencer homomorphism for the deformation determines a section of N, and hence a class $S\frac{!}{\mathfrak{F}}\left[ C \right]$ in $S\frac{{ - 1}}{\mathfrak{F}}\left( C \right)$ which also refines X·V. In fact

$$ \begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cap {V_t}} \right) \subset s\frac{{ - 1}}{\mathfrak{X}}\left( C \right) \subset W $$

and, by these inclusions

$$ \begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right) \to s\frac{!}{\mathfrak{K}}\left[ C \right] \to X \cdot V \cdot $$

If X _t , meets V_t properly for generic t, then $\begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cap {V_t}} \right)$ has dimension k - d, so $\begin{array}{*{20}{c}}dim {\lim} \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right)$ is a well-defined cycle representing X·V. If dim $s\frac{{ - 1}}{\mathfrak{K}}\left( C \right) = k - d$ this limit cycle must be $s\frac{!}{\mathfrak{K}}\left[ C \right]$ in which case the limit cycle is determined by infinitesimal data.

This allows a dynamic interpretation for the distinguished varieties and their equivalences, which can be useful for calculations. For any closed subset Z of X, let (X·V)^Z be the part of X·V supported on Z (§ 6.1). If N _X Y is generated by its sections, there is an open set T (Z) of sections such that for each, s ^! [C] is a (k - d)-cycle $\begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right)$ and the part of s ^! [C] which is supported on Z is precisely (X·V) ^Z Thus (X·V) ^Z is represented by the part of the limit cycle supported on Z, for generic deformations, i.e., deformations whose characteristic section is in T (Z). Knowing (X·V) ^Z for all Z is the same as knowing the equivalences of the distinguished varieties.