Perverse Filtrations via Brylinski-Radon transformations

TL;DR

This paper proves the t-exactness of Brylinski-Radon transformations on sheaves over flag varieties, leading to new Lefschetz-type results, sharpening of divisibility theorems, and an upgraded functor equivalence.

Contribution

It establishes the t-exactness of Brylinski-Radon transformations and derives several significant cohomological and functorial results from this.

Findings

Proves t-exactness of Brylinski-Radon transformation.

Derives weak Lefschetz type results for cohomology.

Sharpened cohomological divisibility theorems and functor equivalences.

Abstract

In this article, we prove the $t$-exactness of a Brylinski-Radon transformation taking values in sheaves on flag varieties. This implies several weak Lefschetz type results for cohomology. In particular, we obtain de Cataldo-Migliorini's P=Dec(F) and Beilinson's basic lemma, the latter was an important ingredient in their proof of P=Dec(F). Our methods also allow the sharpening of Esnault-Katz's cohomological divisibility theorem and estimates for the Hodge level. Finally, we upgrade P=Dec(F) to an equivalence of functors which is also valid over a base.