Systolic inequalities for K3 surfaces via stability conditions

Abstract We introduce the notions of categorical systoles and volumes Bridgeland stability conditions on triangulated categories. prove that for any projective K3 surface X , there exists a constant C depending only rank discriminant NS ( ), such $$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot {vol}(\sigma ) \end{aligned}$$ sys ( ? ) 2 ? C · vol holds condition $$\mathcal {D}^b\mathrm {Coh}(X)$$ D b Coh X . This is an algebro-geometric generalization classical systolic inequality two-tori. also discuss applications this in symplectic geometry.