Differentiability of restricted volumes of big divisors

Question

Let $V$ be a subvariety of a smooth projective variety $X$ of dimension $n$. The restricted volume on $V$ of a $\mathbb{Q}$--divisor $D$ on $X$ is defined by $$\mathrm{Vol}_{X|V}(D):=\limsup_{k\to \infty}\frac{d!}{k^d}h^0(X|V, kD) $$ where $d=\dim V$ and $h^0(X|V, kD)$ is the dimension of the image of the restriction map $$H^0(X, kD)\to H^0(V, kD|_V).$$

It is well known that the volume function $\mathrm{Vol}_X: \mathrm{Big}(X)\to \mathbb{R}_{\geq 0}$ is a differentiable function in any direction, that is $\mathrm{Vol}_X(D+tE)$ is differentiable at $t=0$ for any big divisor $D$ and effective $E$.

Is the restricted volume function also differentiable, i.e. is $\mathrm{Vol}_{X|V}(D+tE)$ differentiable at $t=0$? Any references or counter examples?

Answer 1

As requested, I am reposting my comment as an answer. Note that Boucksom, Favre, and Jonsson only state that $X = \operatorname{Bl}_P\mathbf{P}^2$ gives an example of a restricted volume that is not $\mathcal{C}^1$. We will show that their example also gives an example of a non-differentiable restricted volume.

Example (see [Boucksom–Favre–Jonsson, Ex. 4.17]). Let $X$ be the blowup of $\mathbf{P}^2$ at a point, and let $h,e \in N^1(X)$ be the class of the pullback of a line and the exceptional class, respectively. We then have the following decomposition of the effective cone of $X$:

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The positive part $P$ of the Zariski decomposition for a point $\alpha = xh-ye$ is as described in the figure above. By [Ein–Lazarsfeld–Mustaţă–Nakamaye–Popa, Ex. 2.19], the restricted volume for the class $\alpha = xh-ye$ along a curve $C$ with class $ah-be$ can be computed as $$\begin{aligned} \operatorname{vol}_{X \vert C}(\alpha) &= P(\alpha)\cdot(ah-be)\\ &= \begin{cases} ax-by & \alpha \in \operatorname{Nef}^1(X),\\ ax & \alpha \in \operatorname{\overline{Eff}}^1(X) \smallsetminus \operatorname{Nef}^1(X). \end{cases} \end{aligned}$$ For $\alpha = h+te$, we have $$\operatorname{vol}_{X \vert C}(h+te) = \begin{cases} a+bt & t\le0,\\ a & t\ge0, \end{cases}$$ which is not differentiable at $t = 0$ if $b \ne 0$.