On degenerations of projective varieties to complexity-one T-varieties
Kiumars Kaveh, Christopher Manon, Takuya Murata

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 be a positively graded finitely generated -domain with Krull dimension . We show that there is a homogeneous valuation of rank such that the associated graded is finitely generated. This then implies that any polarized -dimensional projective variety has a flat deformation over , with reduced and irreducible fibers, to a polarized projective complexity-one -variety (i.e. a variety with a faithful action of a -dimensional torus ). As an application we conclude that any -dimensional complex smooth projective variety equipped with an integral K\"ahler form has a proper -dimensional Hamiltonian torus action on an open dense subset that extends continuously to all of .
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Nonlinear Waves and Solitons
