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

Abstract

We study semi-algebraic domains associated with symplectic tori and conjecturally identified with spaces of stability conditions on the Fukaya categories of these tori. Our motivation is to test which results from the theory of flat surfaces could hold for more general spaces of stability conditions. The main results concern systolic bounds and volume of the moduli space.

Extremal length in higher dimensions and complex systolic inequalities

“…The following result provides a simple case in which extremal volume can be used to establish complex systolic inequalities. A similar result, in a different context, appears in [6].…”

“…Within our framework, the simplest definition would be: given (J, Ω), the (J, Ω)-systole of M is the smallest non-zero value of L Ω L amoung all L representing any non-zero class in H n (M, Z). This definition has the positive feature that it depends only on the equivalence class of Ω, allowing for comparisons with extremal volume as in Loewner's 1-dimensional case (6).…”

“…Jason Lotay. While thinking about geometric inequalities related to extremal volume and Theorem 6.4, I came across [5], which defines a more restrictive notion of systoles for Calabi-Yau manifolds, and [6], which proves essentially the same result as Theorem 6.4, but with no relation to extremal length and with a focus on symplectic rather than complex geometry. I thus realized that systolic geometry provides a natural context for such inequalities.…”

Systolic inequalities for K3 surfaces via stability conditions

“…After the first version of the present article was posted online, Haiden [13] proves a systolic inequality for certain higher-dimensional symplectic torus, and Pacini [25] proposes a higherdimensional generalization of extremal lengths and complex systolic inequalities.…”

“…Proof The first three statements follow easily from the definition. To prove the last statement, we use a similar idea in the proof of [13,Theorem 4.3]. One can verify by direct computation that…”