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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

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

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$.

Christoph Lossen