Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On degenerations of projective varieties to complexity-one T-varieties   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 $X$.

24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat