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arXiv · 2507.06816

Some general \'etale Weak Lefschetz-type theorems

Sergei I. Arkhipov·Mikhail V. Bondarko
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Abstract

We establish new general etale versions of theorems of Barth and Sommese. Respectively, we compute the lower etale cohomology of closed subvarieties of $P^N$ of small codimensions and of their preimages with respect to proper morphisms (that are not necessarily finite; this statement is completely new), and also of the zero loci of sections of ample vector bundles; all these statements are valid over fields of arbitrary characteristics. To obtain these results, we use a new 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties that is related to a result of Goresky and MacPherson (over complex numbers).

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Sergei I. Arkhipov, Mikhail V. Bondarko. 2025-07-09. Some general \'etale Weak Lefschetz-type theorems. https://arxiv.org/abs/2507.06816

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