Namespaces
Variants
Actions

Difference between revisions of "Gauss-Manin connection"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (fix tex)
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
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]]
+
<!--
 +
g0434701.png
 +
$#A+1 = 52 n = 1
 +
$#C+1 = 52 : ~/encyclopedia/old_files/data/G043/G.0403470 Gauss\ANDManin connection
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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 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|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|Differential]]). Each derivation  $  \theta $
 +
of  $  K $(
 +
cf. [[Derivation in a ring|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) $[[#References|[a1]]], [[#References|[a2]]]. This amounts to a [[Connection|connection]]
  
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
+
$$
 +
\nabla : \
 +
H _ { \mathop{\rm dR}  }  ^ {1} ( X/K)  \rightarrow \
 +
\Omega _ {K}  ^ {1} \otimes H _ { \mathop{\rm dR}  }  ^ {1} ( X/K)
 +
$$
  
<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>
+
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|Regular singular point]]).
  
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" />).
+
The generalization to higher dimension is due to A. Grothendieck [[#References|[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
  
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
+
$$
 +
H _ { \mathop{\rm dR}  }  ^ {n}
 +
( {X  ^ {h} } / {S  ^ {h} } )  \cong \
 +
{\mathcal O} _ {S  ^ {h}  } \otimes _ {\mathbf C} R  ^ {n} f _  \star  \mathbf C _ {X  ^ {h}  } ,
 +
$$
  
<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>
+
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}  } $).
  
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
+
An algebraic construction has been given by N.M. Katz and T. Oda [[#References|[a4]]]. The complex  $  \Omega _ {X/ \mathbf C }  ^ {\bullet } $
 +
is filtered by subcomplexes  $  \phi  ^ {i} $,
 +
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/g04347036.png" /></td> </tr></table>
+
$$
 +
\phi  ^ {i} \Omega _ {X/ \mathbf C }  ^ {p}  = \
 +
\textrm{ image }  \textrm{ of } \
 +
( f ^ { \star } \Omega _ {S/ \mathbf C }  ^ {i} \otimes
 +
\Omega _ {X/ \mathbf C }  ^ {p - i }  \rightarrow \
 +
\Omega _ {X/ \mathbf C }  ^ {p} ).
 +
$$
 +
 
 +
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.
 
is an algebraic version of the Gauss–Manin connection.
Line 23: Line 84:
 
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]]].
 
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
+
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
  
<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>
+
$$
 +
[ 0 \rightarrow \dots \rightarrow  0  \rightarrow \
 +
\Omega _ {X/S}  ^ {p}  \rightarrow \
 +
\Omega _ {X/S} ^ {p + 1 }  \rightarrow \dots ]
 +
$$
  
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
+
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} $[[#References|[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
  
<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>
+
$$
 +
\nabla ( F ^ { p } H _ { \mathop{\rm dR}  }  ^ {n} ( X/S))  \subseteq \
 +
\Omega _ {S}  ^ {1} \otimes F ^ { p - 1 } H _ { \mathop{\rm dR}  }  ^ {n} ( X/S).
 +
$$
  
The geometric data <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 geometric data $  ( H _ { \mathop{\rm dR}  }  ^ {n} ( X/S), \nabla , F  ) $
 +
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.
 
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]]].
+
In the theory of $  D $-
 +
modules (cf. [[D-module| $  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|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====
 
====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</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)</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</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</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)</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>
+
<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>

Latest revision as of 11:55, 12 January 2021


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

[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. , (1968) pp. 199–213
[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
How to Cite This Entry:
Gauss-Manin connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss-Manin_connection&oldid=22497