User:Paczki sa nalepsze z dzemem/sandbox

Definition

Consider a holomorphic complex function germ

$f:(\mathbb {C} ^{n},0)\to (\mathbb {C} ,0)\$

and denote by ${\mathcal {O}}_{n}$ the ring of all function germs $(\mathbb {C} ^{n},0)\to (\mathbb {C} ,0)$. Every level of a function is a complex hypersurface in $\mathbb {C} ^{n}$, therefore we will call $f$ a hypersurface singularity.

Assume it is an isolated singularity: in case of holomorphic mappings we say that a hypersurface singularity $f$ is singular at $0\in \mathbb {C} ^{n}$ if its gradient $\nabla f$ is zero at $0$ and that a singular point is isolated if it is the only singular point in a sufficiently small neighbourhood. In particular, the multiplicity of the gradient

$\mu (f)=\dim _{\mathbb {C} }{\mathcal {O}}_{n}/\nabla f$

is finite. This number $\mu (f)$ is the Milnor number of singularity $f$ at $0$.

Geometric interpretation

Milnor originally introduced $\mu (f)$ in geometric terms in the following way. All fibers $f^{-1}(c)$ for values $c$ close to $0$ are nonsingular manifolds of real dimension $2(n-1)$. Their intersection with a small open disc $D_{\epsilon }$ centered at $0$ is a smooth manifold $F$ called the Milnor fiber. Up to diffeomorphism $F$ does not depend on $c$ or $\epsilon$ if they are small enough. It is also diffeomorphic to the fiber of the Milnor fibration map.

The Milnor fiber $F$ is a smooth manifold of dimension $2(n-1)$ and has the same homotopy type as a bouquet of $\mu (f)$ spheres $S^{n-1}$. This is to say that its middle Betti number $b_{n-1}(F)$ is equal to the Milnor number and it has homology of a point in dimension less than $n-1$. For example, a complex plane curve near every singular point $z_{0}$ has its Milnor fiber homotopic to a wedge of $\mu _{z_{0}}(f)$ circles (Milnor number is a local property, so it can have different values at different singular points).

Thus we have equalities

Milnor number = number of spheres in the wedge = middle Betti number of $F$ = degree of the map $z\to {\frac {{\nabla }f(z)}{\|{\nabla }f(z)\|}}$ on $S_{\epsilon }$ = multiplicity of the gradient $\nabla f$

Another way of looking at Milnor number is by perturbation. We say that a point is a degenerate singular point, or that f has a degenerate singularity, at $z_{0}\in \mathbb {C} ^{n}$ if $z_{0}$ is a singular point and the Hessian matrix of all second order partial derivatives has zero determinant at $z_{0}$:

$\det \left({\frac {\partial ^{2}f}{\partial z_{i}\partial z_{j}}}\right)_{1\leq i\leq j\leq n}^{z=z_{0}}=0.$

We assume that f has a degenerate singularity at 0. We can speak about the multiplicity of this degenerate singularity by thinking about how many points are infinitesimally glued. If we now perturb the image of f in a certain stable way the isolated degenerate singularity at 0 will split up into other isolated singularities which are non-degenerate! The number of such isolated non-degenerate singularities will be the number of points that have been infinitesimally glued.

Precisely, we take another function germ g which is non-singular at the origin and consider the new function germ h := f + εg where ε is very small. When ε = 0 then h = f. The function h is called the morsification of f. It is very difficult to compute the singularities of h, and indeed it may be computationally impossible. This number of points that have been infinitesimally glued, this local multiplicity of f, is exactly the Milnor number of f.

Further contributions give meaning to Milnor number in terms of dimension of the space of versal deformations, i.e. the Milnor number is the minimal dimension of parameter space of deformations that carry all information about initial singularity.