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

Let R be a positively graded finitely generated k-domain with Krull dimension d + 1. We show that there is a homogeneous valuation v: R \ {0} → ℤ^d of rank d such that the associated graded gr_v(R) is finitely generated. This then implies that any polarized d-dimensional projective variety X has a flat deformation over A¹, 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.