Valuative stability of polarised varieties

Abstract

Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to Kstability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita's βinvariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the δ-invariant plays in the study of valuative stability and K-stability of polarised varieties.

Openness of uniformly valuative stability on the Kähler cone of projective manifolds

“…In [19], Dervan and Legendre define the new β-invariant for general polarised varieties by computing the Donaldson-Futaki invariant of the test configuration associated to a dreamy divisor, then show that valuative stability for dreamy divisors is equivalent to K-stability for integral test configurations. Here an integral test configuration means that its central fiber is integral.…”

“…Here an integral test configuration means that its central fiber is integral. In this paper, we do not involve the explicit definition of K-stability, refer to [19], [12], [21].…”

“…In this paper, we establish the openness of uniformly valuative stability (see Definition 3.5, which is stronger than uniformly valuative stability given by Dervan and Legendre in [19]) for general projective varieties. Our main theorem is Theorem 1.1 (see Theorem 3.7).…”

“…For any valuation or divisorial valuation, one can associate a filtration to this valuation, when this filtration is finite generated, such a valuation is called a dreamy valuation or dreamy divisor. Dervan and Legendre [19] define a new β-invariant for polarised varieties, which generalizes the original Fujita's β-invariant, by computing the Donaldson-Futaki invariant of the test configuration associated a dreamy divisor.…”

“…

Assume that a projective variety with a polarization is uniformly valuatively stable (stronger than uniformly valuatively stable given by Dervan and Legendre in [19]), then the projective variety with any polarization sufficiently close to the original polarization is uniformly valuatively stable. We also define valuative stability for the transcendental Kähler classes.

…”