Openness of uniformly valuative stability on the Kahler cone of projective manifolds

Assume that a projective variety is uniformly valuatively stable with respect to a polarization. We show that the projective variety is uniformly valuatively stable with respect to any polarization sufficiently close to the original polarization. The definition of uniformly valuatively stability in this paper is stronger than that given by Dervan and Legendre (Valuative stability of polarised varieties, arXiv:2010.04023, 2020). We also define the valuative stability for the transcendental Kahler classes. Our openness result can be extended to the Kahler cone of projective manifolds.

Automated summary

Assuming that a projective variety is uniformly valuatively stable with respect to a polarization, we prove that it remains uniformly valuatively stable with respect to any polarization sufficiently close to the original one. The definition of uniformly valuatively stability in this paper is stronger than that proposed by Dervan and Legendre (Valuative stability of polarized varieties, arXiv:2010.04023, 2020). We also introduce the notion of valuative stability for transcendental Kahler classes. Our openness result can be extended to the Kahler cone of projective manifolds.