Stability Functions

Abstract

Abstract. In this article we discuss the role of stability functions in geometric invariant theory and apply stability function techniques to problems in toric geometry. In particular we show how one can use these techniques to recover results of [BGU] and [STZ] on asymptotic properties of sections of holomorphic line bundles over toric varieties.

Asymptotics of eigensections on toric varieties

“…In [3] the smooth case is also considered. There the authors take advantage of the Delzant construction which realizes X as a certain GIT-quotient of the set of stable points of a linear torus action on C d .…”

Kähler quantization and entanglement

“…Remark 2.4. Equivalently, the Bohr-Sommerfeld condition may be stated as a requirement that L Λ has a nonzero covariant constant section [10,20].…”

“…Associating semiclassical states to Lagrangian submanifolds of a symplectic manifold is an idea that arises in many contexts. A partial list of references where various versions of this appear includes [5,8,10,12,17,20,25] and three papers (co-)authored by the first author of this paper: [2,15,16].…”

Stability and integration over Bergman metrics

“…It is also natural to choose S k to be a stability function or a functional closely related to stability. We review the notion of stability in the following section 2, and refer to [4,11,13,21,31,37] for several expositions of GIT stability.…”

“…For convergence of the integral, the y-intercept is also important. Aside from this article, they are studied in [4] but only in the case where f is compactly supported. For the partition function, f ≡ 1 and the difficult problem is to determine the λ for which the integral converges.…”