If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a nonnegative integer, or is infinite. It can be considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory.
Assume it is an isolated singularity: in case of holomorphic mappings we say that a hypersurface singularity is singular at if its gradient is zero at . A singular point is isolated if it is the only singular point in a sufficiently small neighbourhood. In particular, the multiplicity of the gradient
is finite. This number is the Milnor number of singularity at .
Milnor originally introduced in geometric terms in the following way. All fibers for values close to are nonsingular manifolds of real dimension . Their intersection with a small open disc centered at is a smooth manifold called the Milnor fiber. Up to diffeomorphism does not depend on or if they are small enough. It is also diffeomorphic to the fiber of the Milnor fibration map.
The Milnor fiber is a smooth manifold of dimension and has the same homotopy type as a bouquet of spheres . This is to say that its middle Betti number is equal to the Milnor number and it has homology of a point in dimension less than . For example, a complex plane curve near every singular point has its Milnor fiber homotopic to a wedge of 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 = degree of the map on = multiplicity of the gradient
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 if is a singular point and the Hessian matrix of all second order partial derivatives has zero determinant at :
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.
Notice that this quotient space will actually be a vector space, although it may not be finite-dimensional. The Milnor number is then equal to the complex dimension of the local algebra:
It follows from Hilbert's Nullstellensatz that is finite if and only if the origin is an isolated critical point of f; that is, there is a neighbourhood of 0 in such that the only critical point of f inside that neighbourhood is at 0.
Here we give some worked examples in two variables. Working with only one is too simple and does not give a feel for the techniques, whereas working with three variables can be quite tricky. Two is a nice number. Also we stick to polynomials. If f is only holomorphic and not a polynomial, then we could have worked with the power series expansion of f.
Consider a function germ with a non-degenerate singularity at 0, say . The Jacobian ideal is just . We next compute the local algebra:
To see why this is true we can use Hadamard's lemma which says that we can write any function as
for some constant k and functions and in (where either or or both may be exactly zero). So, modulo functional multiples of x and y, we can write h as a constant. The space of constant functions is spanned by 1, hence
It follows that μ(f) = 1. It is easy to check that for any function germ g with a non-degenerate singularity at 0 we get μ(g) = 1.
Note that applying this method to a non-singular function germ g we get μ(g) = 0.
Let , then
So in this case .
One can show that if then
This can be explained by the fact that f is singular at every point of the x-axis.
Let f have finite Milnor number μ, and let be a basis for the local algebra, considered as a vector space. Then a miniversal deformation of f is given by
We can collect function germs together to construct equivalence classes. One standard equivalence is A-equivalence. We say that two function germs are A-equivalent if there exist diffeomorphism germs and such that : there exists a diffeomorphic change of variable in both domain and range which takes f to g.
The Milnor number does not offer a complete invariant for function germs. We do have that if f and g are A-equivalent then μ(f) = μ(g). The converse is false: there exist function germs f and g with μ(f) = μ(g) which are not A-equivalent. To see this consider and . We have but f and g are clearly not A-equivalent since the Hessian matrix of f is equal to zero while that of g is not (and the rank of the Hessian is an A-invariant, as is easy to see).
- Arnold, V.I.; Gusein-Zade, S.M.; Varchenko, A.N. (1985). Singularities of differentiable maps. Volume 1. Birkhäuser.
- Gibson, Christopher G. (1979). Singular Points of Smooth Mappings. Research Notes in Mathematics. Pitman.
- Milnor, John (1963). Morse Theory. Annals of Mathematics Studies. Princeton University Press.
- Milnor, John (1969). Singular points of Complex Hypersurfaces. Annals of Mathematics Studies. Princeton University Press.