Return to 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? Any references or counter examples?

I am mainly interested in the case that $\pi: X\to Y$ is a blowing up with the exceptional divisor $E$ and $D$ big. In other words, is $\mathrm{Vol}_{X|E}(D+tE)$ differentiable at $t=0$?

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?

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? Any references or counter examples?

I am mainly interested in the case that $\pi: X\to Y$ is a blowing up with the exceptional divisor $V$ and $D=\pi^*B-tE$ big. In other words, is $\mathrm{Vol}_{X|E}(\pi^*B-tE)$ is differentiable in $t$ over its domain?

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? Any references or counter examples?

I am mainly interested in the case that $\pi: X\to Y$ is a blowing up with the exceptional divisor $E$ and $D$ big. In other words, is $\mathrm{Vol}_{X|E}(D+tE)$ differentiable at $t=0$?

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

Is the restricted volume function also differentiable? Any references or counter examples?

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? Any references or counter examples?

I am mainly interested in the case that $\pi: X\to Y$ is a blowing up with the exceptional divisor $V$ and $D=\pi^*B-tE$ big. In other words, is $\mathrm{Vol}_{X|E}(\pi^*B-tE)$ is differentiable in $t$ over its domain?