ag.algebraic geometry - Localization of vanishing cycles

An answer to this question was given to me by Pierre Schapira. This is known as the microlocal Bertini-Sard theorem (cf. Sheaves on manifolds cor. 8.3.12).

Consider a map $f:Xto A^1$. It induces $f_pi : Xtimes_{A^1} T^*A^1 to T^*A^1$ and $f_d : Xtimes_{A^1} T^*A^1 to T^*X$. Set $Lambda = SS(M)$ the characteristic variety of $M$. This is a closed conic isotropic subset of $T^*X$. Now
$$
supp(phi_{f-t}(M)) subset [ x ~|~ f(x) = t,~(x,df(x)) in Lambda ]
$$
so
$$
[ tin A^1 ~|~ phi_{f-lambda}(M) neq 0 ] subset
[ tin A^1 ~|~ (t,dt)in f_pi f_d^{-1}(Lambda) ]
$$

Now assume that $f$ is compactifiable as $Xoverset{j}{to} bar{X} overset{bar{f}}{to} A^1$, $j$ an open immersion and $bar{f}$ proper. The closure $bar{Lambda}$ of $Lambda$ is $T^*bar{X}$ is a closed conic isotropic subset and since $bar{f}$ is proper, $bar{f}_pi bar{f}_d^{-1}(bar{Lambda})$ is a closed conic isotropic subset of $T^*A^1$. So its intersection with the nowhere vanishing section

$$
[t in A^1 ~|~ (t,dt) in bar{f}_pi bar{f}_d^{-1}(bar{Lambda})]
$$

has dimension 0. Since $f_pi f_d^{-1}(Lambda) subset bar{f}_pi bar{f}_d^{-1}(bar{Lambda})$ the same is true for

$$
[ tin A^1 ~|~ phi_{f-lambda}(M) neq 0 ] subset
[ tin A^1 ~|~ (t,dt)in f_pi f_d^{-1}(Lambda) ]
$$

and the theorem is proved.

If $f$ is algebraic it is always compactifiable. If $f$ is analytic, I don't know if the theorem still holds in general.

PS: If $f(x) = lambda $, the condition $(x,df(x)) in T^*_Z X$ just says that the fiber ${f = lambda}$ is tranverse to $Z$ at $x$. So when $SS(M) subset bigcup T^*_{S_alpha} X$ this gives the geometric interpretation that the vanishing cycles are 0 whenever the fibers are transverse to the strata.