Positivity of supercritical LYZ pairs

We introduce the roots of supercritical Leung–Yau–Zaslow pairs and prove pointwise strong positivity of the associated Schur forms. The resulting LYZ cones contain the pliant cone on smooth projective threefolds, with strict inclusion on a Fujino–Sato threefold. We also establish Hard Lefschetz and Hodge–Riemann relations for the associated Chern classes on compact Kähler manifolds in bidegrees $1\le p+q\le \dim X$ with $\min \{p,q\}\le 1$.