Multidimensional Residues and Applications

  • L. A. Aizenberg
  • A. K. Tsikh
  • A. P. Yuzhakov
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 8)


One of the problems in the theory of multidimensional residues is the problem of studying and computing integrals of the form
$$\int_\gamma {\omega ,} $$
where ω is a closed differential form of degree p on a complex analytic manifold X with a singularity on an analytic set SX, and where γ is a compact p-dimensional cycle in X\S. A special case of this problem is computing the integral (1) when ω is a holomorphic (meromorphic) form of degree p = n = dimc X; in local coordinates the form can be written as ω = f(z) dz = f(z 1, ...,z n )dz 1 ∧ ... ∧ dz n , where f is a holomorphic (meromorphic) function. According to the Stokes formula, the integral (1) depends only on the homology class1 [γ] ∈ H p (X\S) and the De Rham cohomology class [ω] ∈ H P (X\S). Thus in integral (1) the cycle γ can be replaced by a cycle γ1 homologous to it (γ1 ~γ) in X\S and the form ω can be replaced by a cohomologous form ω11 ~ ω) which may perhaps be simpler; for example, it could have poles of first order on S (see § 1, Subsection 4). If γj is a basis for the p-dimensional homology of the manifold X\S, then by Stokes formula for any compact cycle γ ∈ Z p (X\S) the integral (1) is equal to
$$\int_\gamma \omega = \sum\limits_j {k_j \int_{y_j } {\omega ,} } $$
where the k j are the coefficients of the cycle γ as a combination of the basis elements γ j , γ ~ Σ j k j γ j. Formula (2) shows that the problem of computing integral (1) can be reduced t
  1. 1)

    studying the homology group H p (X\S) (finding its dimension and a basis);

  2. 2)

    determining the coefficients of the cycle γ with respect to a basis;

  3. 3)

    computing the integrals over the cycles in the basis.



Complex Manifold Homology Group Transformation Formula Newton Polygon Newton Polyhedron 
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© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • L. A. Aizenberg
  • A. K. Tsikh
  • A. P. Yuzhakov

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