Prym varieties and the geodesic flow on \(\mathrm{SO}(n)\)
The bigonal construction, which associates to any (arbitrarily ramified) double cover of a hyperelliptic curve \(K\to^{\pi}K_ o\to^{\pi_ 0}\mathbb P^ 1\) a similar tower \(C\to C_ 0\to\mathbb P^ 1\) where the points of \(C\) are the different ways of lifting a fibre of \(\pi_ 0\) to a pair of points in \(K\), is studied. \textit{R. Donagi's} tetragonal construction [Bull. Am. Math. Soc., New Ser. 4, 181--185 (1981; Zbl 0491.14016)] is used to show that the Prym varieties \(\text{Prym}(K)\), \(\text{Prym}(C)\) are isogenous; a more detailed analysis proves that they are dual abelian varieties. The map linearizing the geodesic flow on \(\mathrm{SO}(n)\) is then studied and the abelian varieties where it is rendered injective are found. Because of the above results on the bigonal construction, this generalizes, (and gives a direct geometric proof of) earlier results by \textit{L. Haine} [Math. Ann. 263, 435--472 (1983; Zbl 0521.58042)] and \textit{M. Adler} and \textit{P. van Moerbeke} [Invent. Math. 67, 297--331 (1982; Zbl 0539.58012)] for the \(\mathrm{SO}(4)\) case.
scientific article; zbMATH DE number 3903614
scientific article