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Zbl 0851.58026
Vanhaecke, Pol
A special case of the Garnier system, $(1,4)$-polarized abelian surfaces and their moduli. (English)
[J] Compos. Math. 92, No.2, 157-203 (1994). ISSN 0010-437X; ISSN 1570-5846/e

The paper studies the integrable system defined by the potential $V_{\alpha \beta} = (q^2_1 + q^2_2)^2 + \alpha q^2_1 + \beta q^2_2$, where $\alpha$ and $\beta$ denote complex constants. It is shown that, whereas $V_{\alpha \alpha}$ is only Liouville integrable, for $\alpha \ne \beta$ the perturbation $V_{\alpha \beta}$ becomes algebraic completely integrable, which means that the invariant manifolds are affine parts of abelian varieties. In this case $V_{\alpha \beta}$ leads to abelian surfaces of type (1,4). The proof heavily uses the results of {\it C. Birkenhake, D. van Straten} and the reviewer [Math. Ann. 285, No. 4, 625-646 (1989; Zbl 0714.14028)] on the geometry of these abelian surfaces and their moduli spaces. Moreover, the explicit affine coordinates provided by the integrable system are used to derive some new results for abelian surfaces of type (1,4), e.g. a compactification of the moduli space or an explicit equation for the curve of genus 2 which can be naturally associated to such an abelian surface. In the final section it is shown that in the degenerate case $V_{\alpha \alpha}$ the affine invariant manifolds are $\bbfC^*$-bundles over an elliptic curve. Summarizing: the paper exploits a new and beautiful case of the interaction between algebraic geometry and integrable systems.
[H.Lange (Erlangen)]

MSC 2000:: *37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.
14K99 Abelian varieties and schemes

Keywords: abelian surfaces of type (1,4); integrable systems

Citations: Zbl 0714.14028

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