# Relation between Milnor fiber and its restriction via vanishing cycles

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!

## 1 Answer

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.

• Thank you for your answer! (Great to have it from the author...) I do not know what the relative polar curve is, could you please include a reference or a further explanation? Also, can the situation you described for a hypersurface be generalized to the case of $X$ a complete intersection in $U$? Jan 14, 2015 at 17:35
• I think the best two references for you, given the questions that you have, are:Lê, D. T. and Saito, K. La constance du nombre de Milnor donne des bonnes stratifications. C.R. Acad. Sci., 277:793–795, 1973. Lê, D. T. Calcul du Nombre de Cycles Evanouissants d’une Hypersurface Complexe. Ann. Inst. Fourier, Grenoble, 23:261–270, 1973. Jan 14, 2015 at 20:32
• To deal with the complete intersection case, you need the machinery in my paper "Enriched Relative Polar Curves and Discriminants” in Contemporary Mathematics, vol. 474, 107-144, “Singularities I: Algebraic and Analytic Aspects; International Conference in honor of the 60th birthday of Lê Dung Tráng, January 8-26, 2007, Cuernavaca, Mexico” (2008). I should comment that, even in the hypersurface case, one can't use arbitrary hypersurfaces and get good results; there is a technical condition which needs to be satisfied. Jan 14, 2015 at 20:44
• Unfortunately I cannot find the two first references you gave me; I really would like to find them, also because it is always a pleasure to read french... Do you know where I can find them? Jan 14, 2015 at 22:01
• I recommend scholar.google.com Jan 16, 2015 at 7:03