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

Kiumars Kaveh, Christopher Manon, Takuya Murata

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2 Scopus citations


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} → ℤd of rank d such that the associated graded grv(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.

Original languageEnglish
Pages (from-to)2665-2697
Number of pages33
JournalInternational Mathematics Research Notices
Issue number3
StatePublished - 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)


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