On the limiting extremal vanishing for configuration spaces

TL;DR

This paper investigates the asymptotic vanishing of extremal cohomology groups in configuration spaces of complex projective spaces, extending previous results to non-positive degrees and proposing a conjecture on higher slopes.

Contribution

It extends the understanding of extremal cohomology vanishing to non-positive degrees and introduces a conjecture on the eventual vanishing of higher slope cohomology groups.

Findings

Extremal cohomology groups for degrees (2m-2)k+i with i in {1,2,3} eventually vanish.

Extremal cohomology groups for non-positive integers i in {-1,-2,0} also eventually vanish.

Identifies families of unstable cohomology groups that do not vanish, depending on points and dimension.

Abstract

We study the limiting behavior of extremal cohomology groups of $k$-points configuration spaces of complex projective spaces of complex dimension $m\geq 4.$ In the previous work, we prove that the extremal cohomology groups of degrees $(2m-2)k+i$ are eventually vanish for each $i\in\{1,2,3\}.$ In this paper, we investigate the extremal cohomology groups for non-positive integers, and show that these cohomology groups are eventually vanish for $i\in\{-1,-2,0\}.$ As an application, we confirm the validity of more general question of Knudsen, Miller and Tosteson for non-positive integers. We give a certain families of unstable cohomology groups, which are not eventually vanish. The degrees of these families of cohomology groups are depend on the number of points and the dimension of projective spaces. We formulate the conjecture that the cohomology groups of higher slopes are eventually vanish.