6 problems

Let $X$ be an arithmetic variety, meaning a regular scheme projective and flat over $\mathbb Z$ whose generic fibre $X_{\mathbb Q}$ is smooth over $\mathbb Q$. Let…

Let $F$ be a number field, $(X_{\eta},L_{\eta})$ a normal polarized projective variety over $F$, and let $(X,L,h_L)$ range over all metrized polarized normal models over…

Let $(\mathcal{X},\mathcal{L})$ be an arithmetic variety as above, and let $X$ be its corresponding Fano manifold. Assume that $X$ admits a unique Kähler–Einstein metric, with volu…

Let $\mathcal{X}$ be a regular projective arithmetic variety of dimension $n$, and let $\overline{\mathcal{L}}$ be an ample line bundle on $\mathcal{X}$. Let … be a subset such tha…

Let $R$ be a regular arithmetic ring, let $B_0$ be a regular integral scheme flat and of finite type over $R$, and let $X_0$ be a smooth projective integral variety of dimension…

Let $X$ be a projective variety over a global field, let $D$ be a divisor on $X$, and let $\overline{D}$ be a metrized divisor. For each integer $n\geq 1$, set $V_n=H^0(X,nD)$ and…