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Residues of higher order and holomorphic vector fields

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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\(C_I = (c_1 )^{i_1 } \cdot (c_2 )^{i_2 } \cdot \cdot \cdot (c_p )^{i_p } ,(i_1 + 2i_2 + \cdot \cdot \cdot + pi_p = p)\), 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 (α 1, ⋯, αp) and (β 1,⋯, β 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 theα 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:

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Lehmann, D. Residues of higher order and holomorphic vector fields. Ann Glob Anal Geom 12, 109–122 (1994). https://doi.org/10.1007/BF02108292

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