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Prym varieties and the geodesic flow on \(\mathrm{SO}(n)\). (English) Zbl 0566.58028

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. 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 L. Haine [Math. Ann. 263, 435–472 (1983; Zbl 0521.58042)] and M. Adler and P. van Moerbeke [Invent. Math. 67, 297–331 (1982; Zbl 0539.58012)] for the \(\mathrm{SO}(4)\) case.
Reviewer: Stefanos Pantazis

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
14K99 Abelian varieties and schemes
14H30 Coverings of curves, fundamental group
37C10 Dynamics induced by flows and semiflows
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References:

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