Affine Cohomological Transforms, Perversity, and Monodromy
نویسندگان
چکیده
It is now nearly a decade since the theories of perverse sheaves and of the /adic Fourier transform came into being and gave fundamental new insights into the behavior of additive character sums over finite fields. Most of these insights were spellings out of the basic fact that the Fourier transform of a perverse sheaf is itself a perverse sheaf, a statement that amounts to a succession of vanishing statements for various compactly supported cohomology groups. This vanishing, applied to input perverse sheaves that are "mixed", is transformed by Deligne's fundamental Weil II results into archimedean estimates for the corresponding sums. In this paper, we show that a wide class of /-adic "affine cohomological transforms" shares with the Fourier transform the fundamental property of carrying perverse sheaves to perverse sheaves. To define these transforms, fix a· field k of finite characteristic p > 0, a prime number / i= p, integers n ;::: 1 and m ;::: 1, an affine k-scheme V of
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