Stability conditions for polarised varieties

Abstract

We introduce an analogue of Bridgeland's stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical Kähler metric, modelled on the constant scalar curvature Kähler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical Kähler metrics in the large volume regime. We also prove a local converse, and explain how these results can be viewed as producing local wall-crossing phenomena. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang-Mills equation.

Special representatives of complexified Kähler classes

“…Dervan [16] also studies a notion of critical Kähler metrics ω which deforms the scalar curvature s(ω) (nearby the large volume limit) with a suitable notion of central charge Z(ω). Accordingly, one may consider extending this to complexified Kähler classes by the equation…”

“…By standard arguments, this can be solved for all sufficiently small τ , provided [ω] admits a cscK representative ω 0 , and (X, [ω]) has discrete automorphisms (this seems closely related to special cases of the deformed cscK equation studied in [16]). Note that the corresponding large volume limit equations, in the sense of (1.13), are trivially solvable, by Λ ω 0 (σω 0 + τ Ric(ω 0 )) = 2σ + τ ŝ s(ω 0 ) = ŝ.…”

Extremal metrics on fibrations

“…Secondly, we will later be interested in the case dim X = 2, which is dealt with in Section 5.3, and in which the radius in B on which we actually have a symplectic form is shrinking with ε. Both of these lead us to be more direct in our arguments, in comparison to [10,Section 3.5] where an equivariant version of the Darboux theorem was applied to simplify the problem.…”

“…From here, the proof of Theorem 5.5 is the same as [10,Section 3.5]. Having constructed approximate solutions to all orders, one uses the quantitive inverse function theorem to obtain genuine solutions.…”