In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space.
Definition Let f ( z 0 , … , z n ) {\displaystyle f(z_{0},\dots ,z_{n})} be a non-constant polynomial function of n + 1 {\displaystyle n+1} complex variables z 0 , … , z n {\displaystyle z_{0},\dots ,z_{n}} where the vanishing locus of
f ( z ) and ∂ f ∂ z i ( z ) {\displaystyle f(z)\ {\text{ and }}\ {\frac {\partial f}{\partial z_{i}}}(z)}
is only at the origin, meaning the associated variety X = V ( f ) {\displaystyle X=V(f)} is not smooth at the origin. Then, for K = X ∩ S ε 2 n + 1 {\displaystyle K=X\cap S_{\varepsilon }^{2n+1}} (a sphere inside C n + 1 {\displaystyle \mathbb {C} ^{n+1}} of radius ε > 0 {\displaystyle \varepsilon >0} ) the Milnor fibrationpg 68 associated to f {\displaystyle f} is defined as the map
ϕ : ( S ε 2 n + 1 ∖ K ) → S 1 sending x ↦ f ( x ) | f ( x ) | {\displaystyle \phi \colon (S_{\varepsilon }^{2n+1}\setminus K)\to S^{1}\ {\text{ sending }}\ x\mapsto {\frac {f(x)}{|f(x)|}}} , which is a locally trivial smooth fibration for sufficiently small ε {\displaystyle \varepsilon } . Originally this was proven as a theorem by Milnor, but was later taken as the definition of a Milnor fibration. Note this is a well defined map since
f ( x ) = | f ( x ) | ⋅ e 2 π i Arg ( f ( x ) ) {\displaystyle f(x)=|f(x)|\cdot e^{2\pi i\operatorname {Arg} (f(x))}} , where Arg ( f ( x ) ) {\displaystyle \operatorname {Arg} (f(x))} is the argument of a complex number.
Historical motivation One of the original motivations for studying such maps was in the study of knots constructed by taking an ε {\displaystyle \varepsilon } -ball around a singular point of a plane curve, which is isomorphic to a real 4-dimensional ball, and looking at the knot inside the boundary, which is a 1-manifold inside of a 3-sphere. Since this concept could be generalized to hypersurfaces with isolated singularities, Milnor introduced the subject and proved his theorem.
In algebraic geometry Another closed related notion in algebraic geometry is the Milnor fiber of an isolated hypersurface singularity. This has a similar setup, where a polynomial f {\displaystyle f} with f = 0 {\displaystyle f=0} having a singularity at the origin, but now the polynomial
f t : C n + 1 → C sending ( z 0 , … , z n ) ↦ f ( z 0 , … , z n ) − t {\displaystyle f_{t}\colon \mathbb {C} ^{n+1}\to \mathbb {C} \ {\text{ sending }}\ (z_{0},\ldots ,z_{n})\mapsto f(z_{0},\ldots ,z_{n})-t}
is considered. Then, the algebraic Milnor fiber is taken as one of the polynomials f t ≠ 0 {\displaystyle f_{t\neq 0}} .
Properties and Theorems
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