Residues of higher order and holomorphic vector fields |
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Authors: | Daniel Lehmann |
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Institution: | (1) GETODIM, CNRS, UA 1407, Université de Montpellier II, Case 051 Place E. Bataillon, 34095 Montpellier cedex, France |
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Abstract: | LetX
1 andX
2 be two holomorphic vector fields on a manifoldV with complex dimensionp. Assume that they have the same singular set . For all
, it is known (after Chern-Bott) that each of the vector fields defines a residual characteristic classC
1(V,X
1)(resp.C
1(V,X
2)) inH
2p
(V, V- ), which is a lift of the usual characteristic classC
1
(V) of the tangent bundle. The differenceC
1
(V,X
2)-C
1
(V,X
1) belongs then to the image of in the exact sequence . In fact, there exists a canonical liftC
1
(V,X
1,X
2) of this difference inH
2p–1(V- ): we will call itthe residual class of order 2 (associated toI, X
1 andX
2). This class is localized near the points whereX
1 andX
2 are colinear: we will explain this precisely in terms of Grothendieck residues. The formula that we obtain can be interpreted as a generalization of the purely algebraic identity, obtained from the general one as a byproduct: where (
1, , p) and (
1, ,
p
) denote two families of non-zero complex numbers, such that all denominators in this formula do not vanish. (This identity corresponds in fact to the case whereX
1 andX
2 are non-degenerate at the same isolated singular point.)If the
i
's (1 i p) depend now differentiably (resp. holomorphically) on a real (resp. complex) parametert then, denoting by the derivative with respect tot, and assuming all numbers lying in a denominator not to be 0, we can deduce from the above identity the following derivation formula:![MediaObjects/10455_2005_BF02108292_f3.jpg](/content/p205j1773715n548/MediaObjects/10455_2005_BF02108292_f3.jpg) |
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Keywords: | Holomorphic vector fields Grothendieck residues higher order |
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