Gauß-Manin connection

There is a natural action of the fundamental group $\Pi_1(T',t)$, $t\in T'$, on $H^n(X_t,\mathbf{C})\cong H^n(X^\infty,\mathbf{C})$. A positively oriented generator of $\Pi_1(T',t)$ operates via the monodromy operator $M$ defined by

$$(Ms)(\tau):=s(\tau+1)$$

for $s\in H^n(X^\infty,\mathbf{C})$ and $\tau\in T^\infty$. Let $M=M_sM_u$ be the decomposition of $M$ into the semisimple part $M_s$ and the unipotent part $M_u$, and set $N:=\log M_u$. By the monodromy theorem [1,24], the eigenvalues of $M_s$ are roots of unity and $N^{n+1}=0$. Let

$$H^n(X^\infty,\mathbf{C})\cong\bigoplus_\lambda H^n(X^\infty,\mathbf{C})_\lambda$$

be the decomposition of $H^n(X^\infty,\mathbf{C})$ into the generalized eigenspaces of $M$, $H^n(X^\infty,\mathbf{C})_\lambda:=Ker\,(M_s-\lambda)$, $\lambda=\exp(-2\pi i\alpha)\,,\;\,\alpha\in\mathbf{Q}$, and let $M_\lambda:=M\big\vert _{H^n(X^\infty,\mathbf{C})_\lambda}$. For $A\in H^n(X^\infty,\mathbf{C})_\lambda$, $\lambda=\exp(-2\pi i\alpha)$,

$$s(A,\alpha)(t)\,:=\,t^{\alpha-\frac{N}{2\pi i}}A(t) \,=\,t^\alpha\exp\Bigl(-\frac{N}{2\pi i}\log t\Bigr)A(t),$$

is monodromy invariant and defines a holomorphic section in $H^n$. The sections $i_*s(A,\alpha)$ span a $\partial_t$-invariant, finitely generated, free $\mathcal{O}_T[t^{-1}]$-submodule $\mathcal{G}\subset i_*\mathcal{H}^n$ of rank $\mu$. Note that the direct image sheaf $i_{\ast}\mathcal{H}$ is in general not finitely generated. The Gauß-Manin connection is the regular ${\mathbf{C}\{t\}}[\partial_t]$-module $\mathcal{G}_0$, the stalk of $\mathcal{G}$ at 0 [1,32].