Relation between Milnor fiber and its restriction via vanishing cycles

Question

I am reading these notes on nearby and vanishing cycles, where an initial assumption is made: the author talks about a complex analytic function $f:X\to \mathbb C$, assuming $X$ is closed in an open subset $U\subset \mathbb C^N$ and $f$ it is the restriction of a function $\tilde f:U\to \mathbb C$. Then some statements are made regarding the cohomology of the Milnor fiber $F_{f,p}$ for $p\in f^{-1}(0)$. The function $\tilde f$ is never mentioned again. So I was wondering:

What is the relation between the sheaf of vanishing cycles $\phi_f$ (living over the singular locus of $X_0=f^{-1}(0)$) and $\phi_{\tilde f}$ (living over the singular locus of $U_0=\tilde f^{-1}(0)$)?

In particular I would like to understand under which assumptions one can conclude that $$\chi_{top}(F_{f,p})=\chi_{top}(F_{\tilde f,p}).$$

Even "stronger" question: Can we say that the reduced cohomologies $$ \mathcal H^i(\phi_f \underline{\mathbb Q}_X)_p\cong\tilde H^i(F_{f,p},\mathbb Q),\qquad \tilde H^i(F_{\tilde f,p},\mathbb Q)\cong\mathcal H^i(\phi_{\tilde f} \underline{\mathbb Q}_U)_p $$ agree?

Thank you for reading!

Answer 1

There is no general relationship between the cohomology of the Milnor fibers of $\tilde f$ and $f$, in this setting. At a point $p\in X_0$, the Milnor fiber of $f$ is the intersection of $X$ and the Milnor fiber of $\tilde f$. Hence, the topologies of the two fibers may be very different.

If $X$ is a hyperplane or hypersurface, the situation is manageable and one can analyze how the topology of the Milnor fibers of $f$ and $\tilde f$ compare, but the equality of the Euler characteristics of the Milnor fibers would be very special. For instance, if $\tilde f$ has a 1-dimensional critical locus $\Sigma\tilde f$ at the origin, and $X$ is a hyperplane $H$ such that $f$ has an isolated critical point at the origin, the the Euler characteristics of the two Milnor fibers would be the same if and only if the relative polar curve of $\tilde f$ with respect to $H$ is empty; this is equivalent to saying that $\Sigma\tilde f$ is smooth at the origin and defines a $\mu$-constant (Milnor number constant) family of isolated critical points.

The standard reason for starting with $\tilde f$, even though one is interested in $f$, is so that one can use conormal techniques and the differential $d\tilde f$ when calculating.