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1 results

  • Real and complex indices of vector fields on complete intersection curves with isolated singularity

  • Oliver Klehn
  • Journal:
    Compositio Mathematica / Volume 141 / Issue 2 / March 2005
    Published online by Cambridge University Press:
    10 February 2005, pp. 525-540
    Print publication:
    March 2005
    • Article
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  • If (V, 0) is an isolated complete intersection singularity and X a holomorphic vector field tangent to V, one can define an index of X, the so-called GSV index, which generalizes the Poincaré–Hopf index. We prove that the GSV index coincides with the dimension of a certain explicitly constructed vector space, if X is deformable in a certain sense and V is a curve. We also give a sufficient algebraic criterion for X to be deformable in this way. If one considers the real analytic case one can also define an index of X which is called the real GSV index. Under the condition that X has the deformation property, we prove a signature formula for the index generalizing the Eisenbud–Levine Theorem.