Local cohomology with Schubert support on compact Hermitian symmetric spaces
Abstract
We consider local cohomology sheaves on compact Hermitian symmetric spaces supported in Schubert varieties, which carry natural structures as mixed Hodge modules. Our main result gives an explicit root-theoretic construction, uniform across Lie types, for the D-simple composition factors and weight filtration on these modules. For Lagrangian and orthogonal Grassmannians, we translate these formulas into combinatorial rules involving strict partitions and shifted Dyck patterns, analogous to existing formulas on ordinary Grassmannians. As an application, we expand upon work of Raicu--Weyman to describe the weight filtration on local cohomology supported in symmetric determinantal and Pfaffian varieties. We also give formulas for the Hodge rational homology level and local cohomological defect of every Schubert variety in a compact Hermitian symmetric space. Finally, we tabulate the local cohomology weight filtrations for all Schubert varieties in the two exceptional Hermitian pairs.
Related
- Geometric functionals of Brownian motion on Hermitian symmetric spaces of non-compact type
- Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties
- Big and nef cohomology classes on compact Kähler spaces
- Yamabe-type problems on compact Hermitian manifolds
- Essential quotients of the cohomology of products of symmetric and alternating groups
BibTeX
@misc{perlman2026local,
title = {{Local cohomology with Schubert support on compact Hermitian symmetric spaces}},
author = {Michael Perlman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22960},
eprint = {2609.22960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22960},
}
RIS
TY - UNPB
TI - Local cohomology with Schubert support on compact Hermitian symmetric spaces
AU - Michael Perlman
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22960
UR - https://arxiv.org/abs/2609.22960
AB - We consider local cohomology sheaves on compact Hermitian symmetric spaces supported in Schubert varieties, which carry natural structures as mixed Hodge modules. Our main result gives an explicit root-theoretic construction, uniform across Lie types, for the D-simple composition factors and weight filtration on these modules. For Lagrangian and orthogonal Grassmannians, we translate these formulas into combinatorial rules involving strict partitions and shifted Dyck patterns, analogous to existing formulas on ordinary Grassmannians. As an application, we expand upon work of Raicu--Weyman to describe the weight filtration on local cohomology supported in symmetric determinantal and Pfaffian varieties. We also give formulas for the Hodge rational homology level and local cohomological defect of every Schubert variety in a compact Hermitian symmetric space. Finally, we tabulate the local cohomology weight filtrations for all Schubert varieties in the two exceptional Hermitian pairs.
ER -