Milnor map

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. The general definition is as follows.

Let ${\displaystyle f(z_{0},\dots ,z_{n})}$ be a non-constant polynomial function of ${\displaystyle n+1}$ complex variables ${\displaystyle z_{0},\dots ,z_{n}}$ such that ${\displaystyle f(0,\dots ,0)=0}$, so that the set ${\displaystyle V_{f}}$ of all complex ${\displaystyle (n+1)}$-vectors ${\displaystyle (z_{0},\dots ,z_{n})}$ with ${\displaystyle f(z_{0},\dots ,z_{n})=0}$ is a complex hypersurface of complex dimension ${\displaystyle n}$ containing the origin of complex ${\displaystyle (n+1)}$-space. (For instance, if ${\displaystyle n=1}$ then ${\displaystyle V_{f}}$ is a complex plane curve containing ${\displaystyle (0,0)}$.) The argument of ${\displaystyle f}$ is the function ${\displaystyle f/|f|}$ mapping the complement of ${\displaystyle V_{f}}$ in complex ${\displaystyle (n+1)}$-space to the unit circle ${\displaystyle S^{1}}$ in C. For any real radius ${\displaystyle r>0}$, the restriction of the argument of ${\displaystyle f}$ to the complement of ${\displaystyle V_{f}}$ in the real ${\displaystyle (2n+1)}$-sphere with center at the origin and radius ${\displaystyle r}$ is the Milnor map of ${\displaystyle f}$ at radius ${\displaystyle r}$.

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