6 problems

Partial Okounkov body conjecture. The collection of $\Delta_{\nu}(\hat{E})$ for linear valuations $\nu$ determines the numerical class of $E$.

Let $X$ be a smooth projective variety of dimension $n$, let $\pi:{\rm Bl}_{r}(X)\rightarrow X$ be the blow-up at $r$ general points $x_{1},\dots,x_{r}$, and let $D$ be a big…

Fix a Borel subgroup $B\subset G$ and a sequence $I=(i_1,\ldots,i_\ell)$ of simple roots defining the Bott–Samelson variety $R_I$. Let $P_v(\mathcal L)$ be the Okounkov body of a l…

Linear-growth characterization. A real valuation $v$ has linear growth if and only if there exists a divisorial valuation $v'$ such that $v\leqslant v'$ at the center of $v$ on som…

Let $(X,L)$ be a polarized projective manifold, let $\mathcal{T}$ be a test configuration for $(X,L)$, and let $\varphi$ be a positive metric on $L$. Phong–Sturm showed that these…

Let $K$ be a number field, let $mathcal{X}$ be an arithmetic variety over $OK$, and let $overline{mathcal{L}}$ be a hermitian line bundle on $mathcal{X}$ whose generic fibre…