PreprintCC BY 4.0arXiv
Euler characteristic and flat cohomology of convex projective $4$-manifolds
Abstract
Let M=Ω/Γ be a closed properly convex real projective $4$-manifold. We prove the Chern-Hopf-Thurston conjecture for this class of manifolds, i.e. χ(M)≥0. The argument transfers Akagawa's double-form identity on the Cheng--Yau cone over Ω to a Bochner identity on M, from which we deduce that Hq(M;ΛpE*)=0 whenever q<p(n+1−p)/(n+1), in arbitrary dimension n, for some flat bundle E naturally attached to M. In dimension four, this implies that the cohomology with coefficients in Λ2E* concentrates in degree two.
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BibTeX
@misc{tamburelli2026euler,
title = {{Euler characteristic and flat cohomology of convex projective $4$-manifolds}},
author = {Andrea Tamburelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31085},
eprint = {2610.31085},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31085},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler characteristic and flat cohomology of convex projective $4$-manifolds
AU - Andrea Tamburelli
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31085
UR - https://arxiv.org/abs/2610.31085
AB - Let M=Ω/Γ be a closed properly convex real projective $4$-manifold. We prove the Chern-Hopf-Thurston conjecture for this class of manifolds, i.e. χ(M)≥0. The argument transfers Akagawa's double-form identity on the Cheng--Yau cone over Ω to a Bochner identity on M, from which we deduce that H^q(M;Λ^pE^*)=0 whenever q<p(n+1−p)/(n+1), in arbitrary dimension n, for some flat bundle E naturally attached to M. In dimension four, this implies that the cohomology with coefficients in Λ^2E^* concentrates in degree two.
ER -