Brieskorn lattice

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

$$s\bigl([\omega]\bigr)(t):=\Bigl[\frac{\omega}{{d\!\,} f}\Big\arrowvert_{X_t}\Bigr]= \Bigl[res\Bigl(\frac{\omega}{f-t}\Bigr)\Bigr]$$

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 $\mathcal{H}''_0\subset V^{>-1}$.