Milnor fibration

Let $f:X\to T$ be a Milnor representative [30] of an isolated hypersurface singularity $f:(\mathbf{C}^{n+1},0) \to (\mathbf{C},0)$ with Milnor number $\mu$. Then

$$\SelectTips{cm}{}\xymatrix {f^{-1}(T')\cap X=:X' \ar[r]^-f & T':=T\setminus\{0\}}$$

is a $C^\infty$ fibre bundle with fibres $X_t:=f^{-1}(t)$, $t\in T'$, of homotopy type of a bouquet of $\mu$ $n$-spheres.

The cohomology bundle $H^n:=\bigcup_{t\in T'}H^n(X_t,\mathbf{C})$ is a flat complex vector bundle on $T'$. Hence, there is a natural flat connection on the sheaf $\mathcal{H}^n$ of holomorphic sections in $H^n$ with covariant derivative $\partial_t:\mathcal{H}^n\to \mathcal{H}^n$. It induces a differential operator $\partial_t$ on $(i_*\mathcal{H}^n)_0$ where $i:T'\hookrightarrow T$ denotes the inclusion.

Let $u:T^\infty \to T$, $\tau\mapsto \exp(2\pi i\tau)$, be the universal covering of $T'$ and $X^\infty:=X'\times_{T'}T^\infty$ the canonical Milnor fibre. Then the natural maps $X_{u(\tau)}\cong X^\infty_\tau \hookrightarrow X^\infty$, $\tau\in T^\infty$, are homotopy equivalences. Hence, $H^n(X^\infty,\mathbf{C})$ can be considered as the space of global flat multivalued sections in $H^n$ and as a trivial complex vector bundle on $T^\infty$.