Proposition 3.16 (00QP) · ScalingStacks

Verified tagged author-source HTML · 1912.02360v1 · cited publication edition alignment unverified.

Proposition 3.16. A convex function $u$ is admissible if and only if the Kähler current defined by the psh function $u\circ\text{Log}$ on $(\mathbb{C}^{*})^{n+1}$ extends to a torus invariant Kähler current on $(\mathbb{P}_{\Delta},[\Delta])$ with continuous local potentials.

00QQ

Proof. (Sketch) Convex functions on $(\mathbb{C}^{*})^{n}$ correspond to torus invariant psh functions via the log map (cf. Lemma 4.3 below). If $u$ is admissible, then near the toric boundary the appropriate local potential $u_{m}$ extends continuously over the boundary piece by the growth asymptote assumption and convexity, and the extension remains psh. Conversely, the asymptotic condition is dictated by the local boundedness of $u_{m}$ near the toric boundary pieces. ∎