2023/08/08 by Hugo C. Botós, Botós, Hugo C.
Mathematics · #51M10 #51M15 #57M50 (Primary) #57R18 (Secondary) #57S30 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2308.04204
openalex publication_date 2023/08/08 · openalex created_date 2023/08/10 · openalex updated_date 2026/07/28
We investigate the relationship between three natural invariants of complex hyperbolic disc orbibundles over oriented and closed hyperbolic 2-orbifolds. These invariants are the Euler characteristic χ of the 2-orbifold, the Euler number e of the disc orbibundle, and the Toledo invariant τ of a faithful representation of the surface group into PU(2,1) attached to the complex hyperbolic structure of the disc orbibundle. Based on previous examples, we conjecture that -3|τ| = 2e+2χ always holds. For complex hyperbolic disc orbibundles over 2-orbifolds derived from quadrangles of bisectors via tessellation, we prove that 3τ= 2e+2χ. Furthermore, we demonstrate that -3|τ| = 2e+2χ holds when a section with no complex tangent planes is present.