On degenerations of projective varieties to complexity-one T-varieties

Let $R$ be a positively graded finitely generated $\mathbf{\text{k}}$-domain with Krull dimension $d+1$. We show that there is a homogeneous valuation $\mathfrak{v}:R\setminus \{0\}\to {\mathbb{Z}}^d$ of rank $d$ such that the associated graded ${\text{gr}}_{\mathfrak{v}}(R)$ is finitely generated. This then implies that any polarized $d$-dimensional projective variety $X$ has a flat deformation over ${\mathbb{A}}^1$, with reduced and irreducible fibers, to a polarized projective complexity-one $T$-variety (i.e. a variety with a faithful action of a $(d-1)$-dimensional torus $T$). As an application we conclude that any $d$-dimensional complex smooth projective variety $X$ equipped with an integral K"ahler form has a proper $(d-1)$-dimensional Hamiltonian torus action on an open dense subset that extends continuously to all of $X$.