Self-intersection of a cycle

Question

If $X$ is a smooth projective variety of dimension $2n$, and $V \subset X$ is a smooth subvariety of dimension $\geq n$, then $V \cdot V$ makes sense as a class as an element of the Chow ring $A^*(V)$. When $V$ is a divisor, it's just the restriction of the normal bundle of $V$. If $V$ has higher codimension, does this cycle have a good geometric interpretation?

Answer 1

Since $V$ is smooth, the interpretation is the same. The intersection $V\cdot V \in A^*(V)$ is $c_d(N_{V/X}) \cap [V]$ where $d$ is the codimension of $V$ in $X$ (i.e. the rank of $N_{V/X}$) and $c_d(N_{V/X})$ is the top chern class of the normal bundle. In the case that $V$ is a divisor, $d = 1$ so $N_{V/X}$ is a line bundle on $V$ and $c_1(N_{V/X}) \in A^*(V)$ is exactly the corresponding divisor class under the identification between line bundles bundles and divisors.

More generally, if $V,W$ are any smooth subvarieties of $X$, then the intersection product $V \cdot W \in A^*(Z)$ where $Z = V \cap W$ can be described as follows. Let $d$ be the codimension of $V$ in $X$ and $d'$ the codimension of $Z$ in $W$. We can restrict the bundle $N_{V/X}$ to $Z$ and consider the quotient bundle $E := N_{V/X}|_Z/N_{Z/W}$. Then

$$ V \cdot W = c_{d - d'}(E) \cap [Z] \in A^*(Z) $$

where $d - d' = \operatorname{rk} E$ so $c_{d-d'}$ is the top Chern class.

This can be pushed even farther when the varieties aren't smooth in which case there is no normal bundle and so we need to use deformations to the normal cone instead. For more look at the first say 6 chapters of Fulton's $\textit{Intersection Theory}$.