Corollary 3.28 (00R6) · ScalingStacks

00R6

Corollary 3.28. In the region $\text{Star}(w_{i})+\mathbb{R}_{\geq 0}w_{i}\subset N_{\mathbb{R}}$ , the directional derivative of the canonical extension $u=u^{**}$ satisfies $\langle w_{i},\nabla u\rangle=1.$ In particular, in this region, for any $m$ with $\langle m,w_{i}\rangle=1$ , the function $u_{m}=u-m$ is constant upon translation in the $w_{i}$ -direction.

00R7

Proof. By Remark 3.21, the $\nabla u$ introduced in the above proof is actually the gradient of the extension $u$ over $N_{\mathbb{R}}$ . By the proof above, we know $\langle\nabla u,w_{i}\rangle=1$ on $\text{Star}(w_{i})\subset\partial\Delta_{\lambda}^{\vee}$ . This directional derivative can only increase as $x\in N_{\mathbb{R}}$ moves in the $w_{1}$ -direction. But $\nabla u\in\Delta$ on $N_{\mathbb{R}}$ since the extension is admissible, so $\langle\nabla u,w_{i}\rangle\leq 1$ everywhere, hence the claim. ∎