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We study semi--algebraic domains associated with symplectic tori andconjecturally identified with spaces of stability conditions on the Fukayacategories of these tori. Our motivation is to test which results from thetheory of flat surfaces could hold for more general spaces of stabilityconditions. The main results concern systolic bounds and volume of the modulispace.

An extension of the Siegel space of complex abelian varieties and conjectures on stability structures