Journal of Algebraic Geometry

Author(s): Terence Gaffney
Journal: J. Algebraic Geom. 18 (2009), 547-574.
Posted: November 17, 2008
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Abstract: We show how the Multiplicity Polar Theorem can be used to calculate invariants which describe an ``isolated singularity''. Examples include the defect of a function, which is related to the Euler obstruction, the index of a differential form, the dimension of the space of vanishing cycles of a sheaf of D-modules

relative to a function

at 0, and a formula for the relative cohomology of the Milnor fiber of

where

has an isolated singularity on a complex analytic set with possibly non-isolated singularities. We apply the result on the defect to refine previous work on the $\mathrm{A}_f$ condition.

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