Relative algebro-geometric stabilities of toric manifolds

In this paper we study the relative Chow and K-stability of toric manifolds. First, we give a criterion for relative K-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford criterion in two ways. One is to consider the maximal torus action and its weight polytope. We obtain a reduction by the strategy of Ono [34], which fits into the relative GIT stability detected by Szécelyhidi. The other way relies on C*-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [13, 36]. In the end, we determine me relative K-stability of all toric Fano threefolds and present counterexamples which are relatively K-stable in the toric sense but which are asymptotically relatively Chow unstable.