# Gauss-Manin connection

The Gauss–Manin connection is a way to differentiate cohomology classes with respect to parameters. Consider a smooth projective curve $X$ over a field $K$. Its first de Rham cohomology group $H _ { \mathop{\rm dR} } ^ {1} ( X/K)$ can be identified with the space of differentials of second kind on $X$ modulo exact differentials (cf. Differential). Each derivation $\theta$ of $K$( cf. Derivation in a ring) can be lifted in a canonical way to a mapping $\nabla _ \theta : H _ { \mathop{\rm dR} } ^ {1} ( X/K) \rightarrow H _ { \mathop{\rm dR} } ^ {1} ( X/K)$ satisfying $\nabla _ \theta ( g \omega ) = g \nabla _ \theta ( \omega ) + \theta ( g) \omega$ for $g \in K$, $\omega \in H _ { \mathop{\rm dR} } ^ {1} ( X/K)$[a1], [a2]. This amounts to a connection

$$\nabla : \ H _ { \mathop{\rm dR} } ^ {1} ( X/K) \rightarrow \ \Omega _ {K} ^ {1} \otimes H _ { \mathop{\rm dR} } ^ {1} ( X/K)$$

which is integrable (i.e. $\nabla _ {[ \theta , \theta ^ \prime ] } = [ \nabla _ \theta , \nabla _ {\theta ^ \prime } ]$). If $K$ is a function field in one variable, one obtains the Picard–Fuchs equation $\nabla \omega = 0$, which has regular singular points (cf. Regular singular point).

The generalization to higher dimension is due to A. Grothendieck [a3]. For a proper and smooth morphism $f: X \rightarrow S$ of $\mathbf C$- schemes the de Rham cohomology of the fibres of $f$ is described by the locally free ${\mathcal O} _ {S}$- modules $H _ { \mathop{\rm dR} } ^ {n} ( X/S) = R ^ {n} f _ \star ( \Omega _ {X/S} ^ {\bullet } )$, the relative de Rham cohomology sheaves. From now on suppose that $S$ is of finite type over $\mathbf C$ and let $X ^ {h}$ and $S ^ {h}$ denote the underlying analytic spaces. Then

$$H _ { \mathop{\rm dR} } ^ {n} ( {X ^ {h} } / {S ^ {h} } ) \cong \ {\mathcal O} _ {S ^ {h} } \otimes _ {\mathbf C} R ^ {n} f _ \star \mathbf C _ {X ^ {h} } ,$$

and the analytic version of the Gauss–Manin connection is defined by $\nabla ( g \omega ) = dg \otimes \omega$ for $g$( respectively, $\omega$) a local section of ${\mathcal O} _ {S ^ {h} }$( respectively, $R ^ {n} f _ \star \mathbf C _ {X ^ {h} }$).

An algebraic construction has been given by N.M. Katz and T. Oda [a4]. The complex $\Omega _ {X/ \mathbf C } ^ {\bullet }$ is filtered by subcomplexes $\phi ^ {i}$, where

$$\phi ^ {i} \Omega _ {X/ \mathbf C } ^ {p} = \ \textrm{ image } \textrm{ of } \ ( f ^ { \star } \Omega _ {S/ \mathbf C } ^ {i} \otimes \Omega _ {X/ \mathbf C } ^ {p - i } \rightarrow \ \Omega _ {X/ \mathbf C } ^ {p} ).$$

One has $( \phi ^ {i} / \phi ^ {i + 1 } ) ^ {n} \cong f ^ { \star } \Omega _ {S/ \mathbf C } ^ {i} \otimes \Omega _ {X/S} ^ {n - i }$ and $R ^ {n} f _ \star ( \phi ^ {i} / \phi ^ {i + 1 } ) \cong \Omega _ {S/ \mathbf C } ^ {i} \otimes H _ { \mathop{\rm dR} } ^ {n - i } ( X/S)$. The connecting homomorphism $\nabla : R ^ {n} f _ \star ( \phi ^ {0} / \phi ^ {1} ) \rightarrow R ^ {n + 1 } f _ \star ( \phi ^ {1} / \phi ^ {2} )$ in the long exact hypercohomology sequence associated to the exact sequence

$$0 \rightarrow \ \phi ^ {1} / \phi ^ {2} \rightarrow \ \phi ^ {0} / \phi ^ {2} \rightarrow \ \phi ^ {0} / \phi ^ {1} \rightarrow 0$$

is an algebraic version of the Gauss–Manin connection.

The Gauss–Manin connection is regular singular [a5][a8]. Its monodromy transformations around points at infinity are quasi-unipotent [a6], [a9], [a10], and bounds on the size of its Jordan blocks are known [a7], [a11]. Geometrical proofs of the monodromy theorem are due to A. Landman [a12], C.H. Clemens [a13] and D.T. Lê [a14].

Another important feature of the Gauss–Manin connection is Griffiths' transversality. The relative de Rham cohomology sheaves of a smooth proper morphism $f: X \rightarrow S$ can be filtered as follows. Let $F ^ { p } \Omega _ {X/S} ^ {\bullet }$ be the subcomplex

$$[ 0 \rightarrow \dots \rightarrow 0 \rightarrow \ \Omega _ {X/S} ^ {p} \rightarrow \ \Omega _ {X/S} ^ {p + 1 } \rightarrow \dots ]$$

of $\Omega _ {X/S} ^ {\bullet }$. Then $\mathop{\rm Gr} _ {F} ^ { p } \Omega _ {X/S} ^ {\bullet } \cong \Omega _ {X/S} ^ {p} [- p]$. The spectral sequence $E _ {1} ^ {pq} = R ^ {q} f _ \star \Omega _ {X/S} ^ {p} \Rightarrow H _ { \mathop{\rm dR} } ^ {p + q } ( X/S)$ degenerates at $E _ {1}$[a15] and $E _ {1} ^ {pq}$ is locally free on $S$. Hence $R ^ {n} f _ \star ( F ^ { p } \Omega _ {X/S} ^ {\bullet } )$ maps injectively to a subsheaf $F ^ { p } H _ { \mathop{\rm dR} } ^ {n} ( X/S)$ of $H _ { \mathop{\rm dR} } ^ {n} ( X/S)$. Griffiths' transversality is the property that

$$\nabla ( F ^ { p } H _ { \mathop{\rm dR} } ^ {n} ( X/S)) \subseteq \ \Omega _ {S} ^ {1} \otimes F ^ { p - 1 } H _ { \mathop{\rm dR} } ^ {n} ( X/S).$$

The geometric data $( H _ { \mathop{\rm dR} } ^ {n} ( X/S), \nabla , F )$ have given rise to the concept of a (polarized) variation of Hodge structure. A. Borel has extended the monodromy theorem to this abstract case ([a16], (6.1)).

The Gauss–Manin connection has also been defined for function germs with isolated singularity [a10] and for mapping germs defining isolated complete intersection singularities [a17]. The monodromy of these connections is the classical Picard–Lefschetz monodromy on the vanishing cohomology.

In the theory of $D$- modules (cf. $D$- module), the theory of the Gauss–Manin connection is expressed as a property of the direct image functor for a proper morphism. Combined with the formalism of vanishing cycle functors [a18] it gives rise to the notion of the Gauss–Manin system [a19]. This plays an important role in the asymptotic Hodge theory of singularities [a20][a22].

#### References

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