Fulton's deformation to the normal cone vs Verdier's

Question

Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone:

Verdier's version: $\tilde{X}_Y^\mathrm{Ver} := \operatorname{Bl}_{Y\times \{0\}}(X\times \Bbb A^1_k) - \operatorname{Bl}_Y(X)$

Fulton's version: $\tilde{X}_Y^\mathrm{Ful} := \operatorname{Bl}_{Y\times \{0\}}(X\times \Bbb P^1_k) - \operatorname{Bl}_Y(X)$

Evidently, $\tilde{X}_Y^\mathrm{Ver}$ is just $\tilde{X}_Y^\mathrm{Ful}$ with the subvariety $X\times \{\infty\}\subseteq (X\times \mathbb P^1_k)-(Y\times \{0\})\subseteq \tilde{X}_Y^\mathrm{Ful}$ removed.

My Question: In what sort of situation is it necessary to use one version and not the other?

Answer 1

The following answer was emailed to me by Claude Sabbah. I have received his permission to post it here.

Whenever you need to get some object on $Y$ from an object existing on the normal cone (in fact normal bundle in your case), you need to proceed by pushforward. As it is better to use a proper pushforward, the version $\tilde{X}_Y^{\mathrm{Ful}}$ is best suited to the question. This is the way Fulton defines Chern classes for example. He needs to use a projection formula that only holds when the fibre of the projection is a projective space, not an affine space.

On the other hand, Verdier is not interested in pushing forward to $Y$. The specialization of a perverse sheaf is defined on the normal bundle, and one recovers the nearby cycles by restricting to a section $Y \times \{1\}$, if it exists. Together with Fourier-Sato transform, this also gives the vanishing cycle functor.