Janus-like algebraic varieties
The authors give a systematic study of a very interesting and important phenomenon in the theory of arithmetic quotient spaces: given two bounded, Hermitian, symmetric domains \(D_1 = SP(4; \mathbb{R})/U (2)\), \(D_2 = U(3,1)/U(3) \times U(1)\) associated to different algebraic groups \(SP (4 ; \mathbb{R})\), \(U(3,1)\), their quotient spaces \(\Gamma_1 \backslash D_1\), \(\Gamma_2 \backslash D_2\) can lead to the same projective variety in their compactification \(\overline {\Gamma_1 \backslash D_1} = \overline {\Gamma_2 \backslash D_2}\). More precisely, the authors prove that there are isomorphisms between the following varieties: \[ \text{(i)} \;\overline {X(1)} \cong \overline {Y(1)}, \quad \text{(ii)} \;\overline {X(2)} \cong \overline {Y (\sqrt {-3})}, \quad \text{(iii)} \;\overline {X(3)} \cong \overline {Y(2)}, \quad \text{(iv)} \overline {X(6)} \cong \overline {Y(2 \sqrt {-3})}. \] Here \(X(-)\), \(Y(-)\) refer to respectively the symplectic and unitary quotient spaces and the decoration 1, 2, 3, 6, \(\sqrt {-3}\), \(2 \sqrt {- 3}\) refer to different levels of congruence subgroups. In the paper these varieties with different arithmetic interpretation are called Janus-like and hence the title of the paper.
scientific article; zbMATH DE number 606995