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| In [[mathematics]], the '''Gauss–Manin connection''', introduced by {{harvs|authorlink=Yuri I. Manin|last=Manin|year=1958}}, is a [[connection (mathematics)|connection]] on a certain [[vector bundle]] over a family of [[algebraic variety|algebraic varieties]]. The base space is taken to be the set of parameters defining the family, and the fibers are taken to be the [[de Rham cohomology]] group <math>H^k_{DR}(V)</math> of the fibers ''V''.
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| Flat sections of the bundle are described by [[differential equation]]s; the best-known of these is the [[Picard–Fuchs equation]], which arises when the family of varieties is taken to be the family of [[elliptic curve]]s. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections.
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| ==Example==
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| A commonly cited example is the [[Dwork construction]] of the [[Picard–Fuchs equation]]. Let
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| :<math>V_\lambda(x,y,z) = x^3+y^3+z^3 - \lambda xyz=0 \;</math>
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| be the [[projective variety]] describing the [[elliptic curve]]. Here, <math>\lambda</math> is a free parameter describing the curve; it is an element of the [[complex projective line]] (the family of hypersurfaces in ''n'' − 1 dimensions of degree ''n'', defined analogously, has been intensively studied in recent years, in connection with the [[modularity theorem]] and its extensions).<ref>http://www.math.princeton.edu/~nmk/dworkfam64.pdf</ref> Thus, the base space of the bundle is taken to be the projective line. For a fixed <math>\lambda</math> in the base space, consider an element <math>\omega_\lambda</math> of the associated de Rham cohomology group
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| :<math>\omega_\lambda \in H^1_{dR}(V_\lambda).</math>
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| Each such element corresponds to a period of the elliptic curve. The cohomology is two-dimensional. The Gauss–Manin connection corresponds to the second-order differential equation
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| :<math>(\lambda^3-27) \frac{\partial^2 \omega_\lambda}{\partial \lambda^2}
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| +3\lambda^2 \frac{\partial \omega_\lambda}{\partial \lambda} + \lambda \omega_\lambda =0. </math>
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| ==D-module explanation==
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| In the more abstract setting of [[D-module]] theory, the existence of such equations is subsumed in a general discussion of the [[direct image of a sheaf|direct image]].
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| ==Equations "arising from geometry"==
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| The whole class of Gauss–Manin connections has been used to try for formulate the concept of differential equations that "arise from geometry". In connection with the [[Grothendieck p-curvature conjecture]], [[Nicholas Katz]] proved that the class of Gauss–Manin connections with algebraic number coefficients satisfies the conjecture. This result is directly connected with the [[Siegel G-function|G-function]] concept of [[transcendence theory]], for meromorphic function solutions. The ''Bombieri-Dwork conjecture'', also attributed to André, which is given in more than one version, postulates a converse direction: solutions as G-functions, or p-curvature nilpotent mod p for almost all p, means an equation "arises from geometry".<ref>http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/42639/1/1296_14.pdf</ref><ref>http://www.ams.org/journals/bull/2007-44-04/S0273-0979-07-01178-0/S0273-0979-07-01178-0.pdf, section 1.4.</ref>
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| ==References==
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| {{Reflist}}
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| *{{eom|id=g/g043470}}
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| *{{Citation | last1=Manin | first1=Ju. I. | title=Algebraic curves over fields with differentiation | url=http://mi.mathnet.ru/eng/izv/v22/i6/p737 | language=Russian | mr=0103889 | year=1958 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=22 | pages=737–756}} English translation in {{Citation | last1=Manin | first1=Ju. I. | title=American mathematical society translations: 22 papers on algebra, number theory and differential geometry | origyear=1958 | url=http://books.google.com/books?id=fZ7ms3db_cMC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | language=Russian | isbn=978-0-8218-1737-7 | mr=0103889 | year=1964 | volume=37 | chapter=Algebraic curves over fields with differentiation | pages=59–78}}
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| {{DEFAULTSORT:Gauss-Manin connection}}
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| [[Category:Algebraic geometry]]
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| [[Category:Connection (mathematics)]]
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