Analytic test configurations and geodesic rays

Abstract

Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle L over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of L. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how this recovers the weak geodesic ray of Phong-Sturm.Theorem 1.2. Suppose that F k,λ is left continuous and decreasing in λ and bounded (see (7.2)). Then there is a well-defined limitwhere the star denotes taking the upper semicontinuous regularization after the limit. Furthermore this limit is maximal except possibly for one critical value of λ, and its Legendre transform is a weak geodesic ray. CONVEX MOTIVATION 5Bedford-Taylor which says that such envelopes are maximal (Theorem 4.10). This is then extended to the case of a test curve of singularities, and in Section 6 we discuss the Legendre transform and prove Theorem 1.1. Following these analytic results, we move on to the algebraic picture. In Section 7 we associate a test curve to a suitable filtration of the coordinate ring of (X, L), and prove Theorem 1.2. We then recall how such filtrations arise from test configurations, and in Section 9 show how this agrees with the construction of Phong and Sturm.

Variational and Non-Archimedean Aspects of the Yau–tian–donaldson Conjecture

“…Let ω be the Fubini-Study metric on X = P 1 , normalized to mass 1. A compact, polar Cantor set K ⊂ P 1 carries a natural probability measure without atoms, and the potential u of this measure with respect to ω is smooth outside K, has zero Lelong numbers and does not belong to E. By [RWN14,Dar17b], u defines a locally bounded weak geodesic ray (u t ) emanating from 0 such that E(u t ) = at with a < 0. However, the corresponding ω-psh function U on X × D has zero Lelong numbers, hence U NA = 0 and E NA (U NA ) = 0. where we have set as before λ = V −1 (K X · L n−1 ).…”

Envelopes with Prescribed Singularities

“…6.1]. In [37,Rmk. 6.11], the authors speculate that ϕ 1 should be in C 1,α loc for all α < 1, and actually prove that it is in C 1,1 loc when X = CP 1 , using either the results of [18] or [10].…”

“…We thus show that this stronger result indeed always holds. As a corollary, one can now apply [37,Cor 6.7] to the Hele-Shaw flow on a general Riemann surface X to see that it is always strictly increasing at at least some fixed positive rate.…”

$L^1$ metric geometry of big cohomology classes

“…Given their importance in the above mentioned applications, we are interested to see how one can construct weak geodesic rays inside (E 1 (X, θ), d 1 ), with the hopes of using them in later investigations involving big cohomology classes. To this end, we point out below that the construction of Ross and Witt Nyström [RWN14] not only generalizes to the big case, but it can be shown that their very flexible method gives all possible weak geodesic rays (with minimal singularity) in a unique manner.…”

“…Section 3 is devoted to the proof of Theorem 1.1. In Section 4 we adapt the concepts of [RWN14] to the big context and prove Thoerem 1.2 and Corollary 1.3.…”