Cohomology characters on the incidence correspondence
Abstract
We investigate the cohomology of line bundles on the incidence correspondence, the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it. In characteristic zero, this cohomology is governed by the Borel-Weil-Bott theorem. In characteristic p>0, however, it becomes considerably subtler, and admits an equivalent reformulation in terms of cohomology tables for divided powers of the cotangent bundle on projective space. Our approach to the problem involves passing to infinitesimal thickenings of the incidence correspondence inside the ambient product of projective spaces. This leads to recursive formulas for the cohomology, generalizing earlier work of Donkin, of Liu, and of Gao-Raicu. We obtain generating functions for cohomology characters, expressed using truncated Schur polynomials and symmetric polynomials encoding the higher structure constants of the Verlinde algebras of SU(2) at levels (p-2) and (2p-2). Along the way, we exploit two important connections with multiplication in the graded Green-Han-Monsky representation ring: one relates this ring to cohomology, and another connects to the Verlinde algebras through the work of Coulembier-Etingof-Ostrik.
Related
BibTeX
@misc{kyomuhangi2026cohomology,
title = {{Cohomology characters on the incidence correspondence}},
author = {Annet Kyomuhangi and Emanuela Marangone and Claudiu Raicu and Ethan Reed},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01371},
eprint = {2609.01371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01371},
}
RIS
TY - UNPB
TI - Cohomology characters on the incidence correspondence
AU - Annet Kyomuhangi
AU - Emanuela Marangone
AU - Claudiu Raicu
AU - Ethan Reed
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01371
UR - https://arxiv.org/abs/2609.01371
AB - We investigate the cohomology of line bundles on the incidence correspondence, the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it. In characteristic zero, this cohomology is governed by the Borel-Weil-Bott theorem. In characteristic p>0, however, it becomes considerably subtler, and admits an equivalent reformulation in terms of cohomology tables for divided powers of the cotangent bundle on projective space. Our approach to the problem involves passing to infinitesimal thickenings of the incidence correspondence inside the ambient product of projective spaces. This leads to recursive formulas for the cohomology, generalizing earlier work of Donkin, of Liu, and of Gao-Raicu. We obtain generating functions for cohomology characters, expressed using truncated Schur polynomials and symmetric polynomials encoding the higher structure constants of the Verlinde algebras of SU(2) at levels (p-2) and (2p-2). Along the way, we exploit two important connections with multiplication in the graded Green-Han-Monsky representation ring: one relates this ring to cohomology, and another connects to the Verlinde algebras through the work of Coulembier-Etingof-Ostrik.
ER -