## Abstract

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 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.

Original language | English |
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Pages (from-to) | 2665-2697 |

Number of pages | 33 |

Journal | International Mathematics Research Notices |

Volume | 2023 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1 2023 |

### Bibliographical note

Funding Information:This work was supported by the National Science Foundation Grant [DMS-1601303 to K.K. and DMS-1500966 to C.M.]; the Simons Foundation Collaboration Grant for Mathematicians [to K.K.]; and the Simons Fellowshio [to K.K.].

Publisher Copyright:

© The Author(s) 2022. Published by Oxford University Press. All rights reserved.

## ASJC Scopus subject areas

- Mathematics (all)