Vanishing Vanishing Cycles

Abstract

If $\Adot$ is a bounded, constructible complex of sheaves on a complex analytic space $X$, and $f:X\to\C$ and $g:X\to\C$ are complex analytic functions, then the iterated vanishing cycles $\phi_g[-1](\phi_f[-1]\Adot)$ are important for a number of reasons. We give a formula for the stalk cohomology $H^*(\phi_g[-1]\phi_f[-1]\Adot)_x$ in terms of relative polar curves, algebra, and the normal Morse data and micro-support of $\Adot$.