View source for Gauss-Manin connection

The Gauss–Manin connection is a way to differentiate cohomology classes with respect to parameters. Consider a smooth projective curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434701.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434702.png" />. Its first [[De Rham cohomology|de Rham cohomology]] group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434703.png" /> can be identified with the space of differentials of second kind on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434704.png" /> modulo exact differentials (cf. [[Differential|Differential]]). Each derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434705.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434706.png" /> (cf. [[Derivation in a ring|Derivation in a ring]]) can be lifted in a canonical way to a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434707.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434708.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g0434709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347010.png" /> [[#References|[a1]]], [[#References|[a2]]]. This amounts to a [[Connection|connection]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347011.png" /></td> </tr></table>

which is integrable (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347012.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347013.png" /> is a function field in one variable, one obtains the Picard–Fuchs equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347014.png" />, which has regular singular points (cf. [[Regular singular point|Regular singular point]]).

The generalization to higher dimension is due to A. Grothendieck [[#References|[a3]]]. For a proper and smooth morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347016.png" />-schemes the de Rham cohomology of the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347017.png" /> is described by the locally free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347018.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347019.png" />, the relative de Rham cohomology sheaves. From now on suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347020.png" /> is of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347021.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347023.png" /> denote the underlying analytic spaces. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347024.png" /></td> </tr></table>

and the analytic version of the Gauss–Manin connection is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347026.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347027.png" />) a local section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347028.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347029.png" />).

An algebraic construction has been given by N.M. Katz and T. Oda [[#References|[a4]]]. The complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347030.png" /> is filtered by subcomplexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347031.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347032.png" /></td> </tr></table>

One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347034.png" />. The connecting homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347035.png" /> in the long exact hypercohomology sequence associated to the exact sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347036.png" /></td> </tr></table>

is an algebraic version of the Gauss–Manin connection.

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

Another important feature of the Gauss–Manin connection is Griffiths' transversality. The relative de Rham cohomology sheaves of a smooth proper morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347037.png" /> can be filtered as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347038.png" /> be the subcomplex

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347039.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347040.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347041.png" />. The spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347042.png" /> degenerates at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347043.png" /> [[#References|[a15]]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347044.png" /> is locally free on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347045.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347046.png" /> maps injectively to a subsheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347048.png" />. Griffiths' transversality is the property that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347049.png" /></td> </tr></table>

The geometric data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347050.png" /> have given rise to the concept of a (polarized) variation of [[Hodge structure|Hodge structure]]. A. Borel has extended the monodromy theorem to this abstract case ([[#References|[a16]]], (6.1)).

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

In the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347051.png" />-modules (cf. [[D-module|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347052.png" />-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|vanishing cycle]] functors [[#References|[a18]]] it gives rise to the notion of the Gauss–Manin system [[#References|[a19]]]. This plays an important role in the asymptotic Hodge theory of singularities [[#References|[a20]]]–[[#References|[a22]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.M. Katz, "On the differential equations satisfied by period matrices" ''Publ. Math. IHES'' , '''35''' (1968) pp. 71–106</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Grothendieck, "On the de Rham cohomology of algebraic varieties" ''Publ. Math. IHES'' , '''29''' (1966) pp. 351–359 {{MR|0199194}} {{ZBL|0145.17602}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> N.M. Katz, T. Oda, "On the differentiation of de Rham cohomology classes with respect to parameters" ''J. Math. Kyoto Univ.'' , '''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043470/g04347053.png" />''' (1968) pp. 199–213</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> N. Nilsson, "Some growth and ramification properties of certain integrals on algebraic manifolds" ''Arkiv för Mat.'' , '''5''' (1963–1965) pp. 527–540</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Deligne, "Equations différentielles à points singuliers réguliers" , ''Lect. notes in math.'' , '''163''' , Springer (1970) {{MR|0417174}} {{ZBL|0244.14004}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> N.M. Katz, "The regularity theorem in algebraic geometry" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars (1971) pp. 437–443</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P.A. Griffiths, "Periods of integrals on algebraic manifolds, I, II" ''Amer. J. Math.'' , '''90''' (1968) pp. 568–626; 805–865</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> A. Grothendieck, "Letter to J.-P. Serre" (5.10.1964)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> E. Brieskorn, "Die Monodromie von isolierten Singularitäten von Hyperflächen" ''Manuscr. Math.'' , '''2''' (1970) pp. 103–161</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> N.M. Katz, "Nilpotent connections and the monodromy theorem. Applications of a result of Turrittin" ''Publ. Math. IHES'' , '''39''' (1971) pp. 175–232</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> A. Landman, "On the Picard–Lefschetz formula for algebraic manifolds acquiring general singularities" , Berkeley (1967) (Thesis)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> C.H. Clemens, "Picard–Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities" ''Trans. Amer. Math. Soc.'' , '''136''' (1969) pp. 93–108 {{MR|0233814}} {{ZBL|0185.51302}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Deligne, "Théorème de Lefschetz et critères de dégénérescence de suites spectrales" ''Publ. Math. IHES'' , '''35''' (1968) pp. 107–126</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" ''Invent. Math.'' , '''22''' (1973) pp. 211–319 {{MR|0382272}} {{ZBL|0278.14003}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> G.-M. Greuel, "Der Gauss–Manin Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten" ''Math. Ann.'' , '''214''' (1975) pp. 235–266</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> F. Pham, "Singularités des systèmes différentiels de Gauss–Manin" , Birkhäuser (1979) {{MR|553954}} {{ZBL|0524.32015}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> M. Saito, "Gauss–Manin system and mixed Hodge structure" ''Proc. Japan Acad. Ser A'' , '''58''' (1982) pp. 29–32</TD></TR></table>