On Degenerations of Projective Varieties to Complexity-One <i>T</i>-Varieties

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