Gauss-Manin connection
The Gauss–Manin connection is a way to differentiate cohomology classes with respect to parameters. Consider a smooth projective curve
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
[a1] | Yu. Manin, "Algebraic curves over fields with differentiation" Transl. Amer. Math. Soc. , 37 (1964) pp. 59–78 Izv. Akad. Nauk. SSSR Ser. Mat. , 22 (1958) pp. 737–756 |
[a2] | N.M. Katz, "On the differential equations satisfied by period matrices" Publ. Math. IHES , 35 (1968) pp. 71–106 |
[a3] | A. Grothendieck, "On the de Rham cohomology of algebraic varieties" Publ. Math. IHES , 29 (1966) pp. 351–359 MR0199194 Zbl 0145.17602 |
[a4] | N.M. Katz, T. Oda, "On the differentiation of de Rham cohomology classes with respect to parameters" J. Math. Kyoto Univ. , ![]() |
[a5] | N. Nilsson, "Some growth and ramification properties of certain integrals on algebraic manifolds" Arkiv för Mat. , 5 (1963–1965) pp. 527–540 |
[a6] | P. Deligne, "Equations différentielles à points singuliers réguliers" , Lect. notes in math. , 163 , Springer (1970) MR0417174 Zbl 0244.14004 |
[a7] | N.M. Katz, "The regularity theorem in algebraic geometry" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 437–443 |
[a8] | P.A. Griffiths, "Periods of integrals on algebraic manifolds, I, II" Amer. J. Math. , 90 (1968) pp. 568–626; 805–865 |
[a9] | A. Grothendieck, "Letter to J.-P. Serre" (5.10.1964) |
[a10] | E. Brieskorn, "Die Monodromie von isolierten Singularitäten von Hyperflächen" Manuscr. Math. , 2 (1970) pp. 103–161 |
[a11] | N.M. Katz, "Nilpotent connections and the monodromy theorem. Applications of a result of Turrittin" Publ. Math. IHES , 39 (1971) pp. 175–232 |
[a12] | A. Landman, "On the Picard–Lefschetz formula for algebraic manifolds acquiring general singularities" , Berkeley (1967) (Thesis) |
[a13] | C.H. Clemens, "Picard–Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities" Trans. Amer. Math. Soc. , 136 (1969) pp. 93–108 MR0233814 Zbl 0185.51302 |
[a14] | D.T. Lê, "The geometry of the monodromy theorem" K.G. Ramanathan (ed.) , C.P. Ramanujam, a tribute , Tata IFR Studies in Math. , 8 , Springer (1978) |
[a15] | P. Deligne, "Théorème de Lefschetz et critères de dégénérescence de suites spectrales" Publ. Math. IHES , 35 (1968) pp. 107–126 |
[a16] | W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" Invent. Math. , 22 (1973) pp. 211–319 MR0382272 Zbl 0278.14003 |
[a17] | G.-M. Greuel, "Der Gauss–Manin Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten" Math. Ann. , 214 (1975) pp. 235–266 |
[a18] | P. Deligne, "Le formalisme des cycles évanescents" A. Grothendieck (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7.II , Lect. notes in math. , 340 , Springer (1973) pp. Exp. XIII |
[a19] | F. Pham, "Singularités des systèmes différentiels de Gauss–Manin" , Birkhäuser (1979) MR553954 Zbl 0524.32015 |
[a20] | J. Scherk, J.H.M. Steenbrink, "On the mixed Hodge structure on the cohomology of the Milnor fibre" Math. Ann. , 271 (1985) pp. 641–665 |
[a21] | A.N. Varchenko, "Asymptotic Hodge structure in the vanishing cohomology" Math USSR Izv. , 18 (1982) pp. 469–512 Izv. Akad. Nauk SSSR , 45 : 3 (1981) pp. 540–591; 688 |
[a22] | M. Saito, "Gauss–Manin system and mixed Hodge structure" Proc. Japan Acad. Ser A , 58 (1982) pp. 29–32 |
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