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## Gauß-Manin connection

There is a natural action of the fundamental group , , on . A positively oriented generator of operates via the monodromy operator defined by

for and . Let be the decomposition of into the semisimple part and the unipotent part , and set . By the monodromy theorem [1,24], the eigenvalues of are roots of unity and . Let

be the decomposition of into the generalized eigenspaces of , , , and let . For , ,

is monodromy invariant and defines a holomorphic section in . The sections span a -invariant, finitely generated, free -submodule of rank . Note that the direct image sheaf is in general not finitely generated. The Gauß-Manin connection is the regular -module , the stalk of at 0 [1,32].

Christoph Lossen
2001-03-21