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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

$\displaystyle (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

$\displaystyle 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)$,

$\displaystyle 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].

Christoph Lossen