PreprintCC BY 4.0arXiv
Rank two cohomology jump loci on projective varieties
Abstract
Let X be a smooth connected complex projective variety. We give a structure theorem for irreducible components of the reduced rank-two cohomology jump loci on X. More precisely, for such a component Z, if the general point of Z is projectively virtually abelian or projectively rigid, then either Z is a symmetric square of an algebraic torus, or a quotient of a torus by an involution, or its normalization Zν is a torus. If the general point is neither, then either Zν is a torus, or an étale quotient of Qν × T where Q is a fixed-determinant, fixed-inertia component of the character variety of some DM-curve.
Related
- Profinite completions and cohomology jump loci
- Total Cohomology of Hilbert schemes of Orbisurfaces I: Reduction to Stacky Loci
- Smooth projective surfaces with bounded cohomology property II
- Igusa stacks and the cohomology of Shimura varieties
- Vanishing theorems for L2 Dolbeault cohomology groups of projective Hartogs triangles
BibTeX
@misc{tran2026rank,
title = {{Rank two cohomology jump loci on projective varieties}},
author = {Quoc-Anh Tran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36251},
eprint = {2610.36251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36251},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rank two cohomology jump loci on projective varieties
AU - Quoc-Anh Tran
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36251
UR - https://arxiv.org/abs/2610.36251
AB - Let X be a smooth connected complex projective variety. We give a structure theorem for irreducible components of the reduced rank-two cohomology jump loci on X. More precisely, for such a component Z, if the general point of Z is projectively virtually abelian or projectively rigid, then either Z is a symmetric square of an algebraic torus, or a quotient of a torus by an involution, or its normalization Z^ν is a torus. If the general point is neither, then either Z^ν is a torus, or an étale quotient of Q^ν × T where Q is a fixed-determinant, fixed-inertia component of the character variety of some DM-curve.
ER -