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Brieskorn lattice

The Brieskorn lattice [1] $ \mathcal{H}'':=f_*\Omega_X^{n+1}/{d\!\,}
$ is a free $ \mathcal{O}_T$-module of rank $ \mu$ [38]. The Gelfand-Leray form

$\displaystyle s\bigl([\omega]\bigr)(t):=\Bigl[\frac{\omega}{{d\!\,}

defines a map $ s:\mathcal{H}''\!\to i_{*}\mathcal{H}^n$ with image in $ \mathcal{G}$, inducing an isomorphism $ \mathcal{H}''\big\vert _{T'}\cong\mathcal{H}^n$, and satisfying $ \partial_t s\bigl([{d\!\,}
f\!\!\:\wedge\eta]\bigr)=s\bigl([{d\!\,}\eta]\bigr)$ by the Leray residue formula [1]. Since $ \mathcal{H}''_0=\Omega_{X,0}^{n+1}/{d\!\,}
f\!\!\:\wedge{d\!\,}\Omega_{X,0}^{n-1}$ is a torsion free $ {\mathbf{C}\{t\}}$-module, $ s:\mathcal{H}''_0 \hookrightarrow
\mathcal{G}_0$ is an inclusion, and we identify $ \mathcal{H}''_0$ with its image in $ \mathcal{G}_0$. By a result of [29], we have $ \mathcal{H}''_0\subset V^{>-1}$.

Christoph Lossen