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

TL;DR

This paper demonstrates that any polarized projective variety can be degenerated into a complexity-one T-variety through a flat deformation, revealing new connections between algebraic and symplectic geometry.

Contribution

It introduces a method to construct flat degenerations of polarized varieties to complexity-one T-varieties using homogeneous valuations.

Findings

Existence of a homogeneous valuation with finitely generated associated graded algebra.

Any polarized projective variety admits a flat degeneration to a complexity-one T-variety.

Smooth projective varieties with an integral Kähler form have dense torus actions extending to the whole variety.

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 $\text{gr}_\mathfrak{v}(R)$ 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 $X$.