On Degenerations of Projective Varieties to Complexity-One T-Varieties

Abstract

Abstract Let $R$ be a positively graded finitely generated $\textbf {k}$-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 $\operatorname {gr}_{\mathfrak {v}}(R)$ 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 Kähler form has a proper $(d-1)$-dimensional Hamiltonian torus action on an open dense subset that extends continuously to all of $X$.