PreprintCC BY-NC-SA 4.0HAL
On possible values of the signature of flat unitary bundles over compact surfaces
Abstract
In this paper, we determine the possible signatures of flat unitary bundles over compact connected oriented surfaces of positive genus with nonempty boundary. More precisely, if Σg,n has genus g≥ 1 and n≥ 1 boundary components, then the signatures arising from representations π1(Σg,n)→U(p,q) are exactly the integers m satisfying |m| ≤ (p+q)(2g+n−2) −(2g−2)|p−q| −min{2,n|p−q|}. Every such integer, including the two extremal values, is realized by a block-diagonal representation whose image is contained, after possibly interchanging p and q, in U(1,1)×min{p,q}×U(|p−q|).
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BibTeX
@misc{kim2026possible,
title = {{On possible values of the signature of flat unitary bundles over compact surfaces}},
author = {Inkang Kim and Pierre Pansu and Xueyuan Wan},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15549},
eprint = {2609.15549},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05748505v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On possible values of the signature of flat unitary bundles over compact surfaces
AU - Inkang Kim
AU - Pierre Pansu
AU - Xueyuan Wan
PY - 2026
DA - 2026/09/14
PB - HAL
DO - 10.48550/arxiv.2609.15549
UR - https://hal.science/hal-05748505v1
AB - In this paper, we determine the possible signatures of flat unitary bundles over compact connected oriented surfaces of positive genus with nonempty boundary. More precisely, if Σ_g,n has genus g≥ 1 and n≥ 1 boundary components, then the signatures arising from representations π_1(Σ_g,n)→U(p,q) are exactly the integers m satisfying |m| ≤ (p+q)(2g+n−2) −(2g−2)|p−q| −min{2,n|p−q|}. Every such integer, including the two extremal values, is realized by a block-diagonal representation whose image is contained, after possibly interchanging p and q, in U(1,1)^×min{p,q}×U(|p−q|).
ER -