Systolic inequalities for K3 surfaces via stability conditions

Abstract

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 sys(σ)2≤C·vol(σ)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathrm {sys}(\\sigma )^2\\le C\\cdot \\mathrm {vol}(\\sigma ) \\end{aligned}$$\\end{document}holds for any stability condition on mathcal {D}^bmathrm {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.