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

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

2003-01-15

arxiv.org/abs/math/0301166v2

Mathematics

Algebraic Geometry

Major revision. Main changes are the corrections of the statements and proofs of the main theorems. 16 pages, to appear in Com

Scientific paper

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 Poincare-Hopf index. We prove that the GSV index coincides with the dimension of a certain explicitely 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.

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