# Gauss-Manin connection

Jump to: navigation, search

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=51288
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9662975668907166, "perplexity": 1349.0703445376237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362619.23/warc/CC-MAIN-20211203091120-20211203121120-00037.warc.gz"}