Holomorphic line bundles on a special abelian surface
Jae-Hyun Yang
TL;DR
This work constructs and analyzes a special abelian surface $A_\Omega$ arising from a new Siegel upper half-space $\widehat{\mathbb H}_2$ and provides explicit descriptions of holomorphic line bundles over $A_\Omega$ via the Appell– Humbert correspondence $L(H,\chi)$. It proves that $L(H_\Omega,\chi_\Omega)$ is ample with first Chern class $E_\Omega={\rm Im}\,H_\Omega$ and gives exact formulas for the dimension of global sections, $\dim H^0(A_\Omega,L(H_\Omega,\chi_\Omega))=\sqrt{\det E_\Omega}=(\operatorname{Im}\tau)^2-(\operatorname{Im}\!z)^2$, while also providing explicit dimensions for other choices $H_\tau$, $H_*$ and their semi-characters. The paper links the analytic model of the new domain to the algebro-geometric structure via the Poincaré bundle, dual abelian variety, and the Picard group, and analyzes the symmetry group $\widehat{G}$ (showing it is not semisimple) and transitive actions on the corresponding homogeneous spaces. Overall, it offers concrete tools for constructing and classifying line bundles on a geometrically distinguished abelian surface, tying together representation-theoretic, geometric, and cohomological perspectives.
Abstract
We consider a special abelian surface $A_Ω$ deduced from the work of Tianze Wang, Tianqin Wang and Hongwen Lu \cite{WWL}. We study holomorphic line bundles over a special abelian surface explicitly.