We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface X, there exists a constant C depending only on the rank and discriminant of NS(X), such that

$$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot \mathrm {vol}(\sigma ) \end{aligned}$$

holds for any stability condition on \(\mathcal {D}^b\mathrm {Coh}(X)\). This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.

中文翻译：

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通过稳定性条件的 K3 表面收缩不等式

我们在三角分类上介绍了布里奇兰稳定条件的分类收缩和分类体积的概念。我们证明对于任何投影 K3 表面X，存在一个常数C仅取决于NS ( X )的秩和判别式，使得

$$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot \mathrm {vol}(\sigma ) \end{aligned}$$

对于\(\mathcal {D}^b\mathrm {Coh}(X)\)上的任何稳定性条件都成立。这是二环上经典收缩不等式的代数几何推广。我们还讨论了这种不等式在辛几何中的应用。