94 problems

Let $K$ be a field complete with respect to a non-archimedean absolute value, let $A/K$ be a semiabelian variety, and let $X \subset A$ be a closed subvariety. Tate–Voloch conjectu…

Let $f(z)\in\mathbb{C}[z_0,\dots,z_{d+1}]$ be a generic homogeneous polynomial of degree $d+2$, where $d\geq1$, and let … be the resulting maximally degenerate one-parameter family…

Let $(X,L)$ be a polarized variety and let $(d4c^1)^{\mathrm{NA}}$ be the space of non-Archimedean finite-energy metrics, with non-Archimedean entropy functional…

Let $(X^{an},\Omega)$ be the $K$-analytic K3 surface with volume form constructed from a chosen set ${\cal L}$ of lines. Independence conjecture. The isomorphism class of the pair…

Let $\widetilde P$ be the complex period homomorphism from the cohomology of the smooth affine base to $({{\bf C}}_t^{mer})^\times$, and let $P'$ be the homomorphism to $K^\times$…

Let $K={{\bf C}}((t))$ and let $X^{an}/K$ be an analytic K3 surface admitting an analytic torus fibration $X^{an}\to S^2$ with standard singularities. Intrinsic period-lattice conj…

Let $\pi:{X}^{an}\to B$ be the continuous map from the analytic Calabi–Yau space to its limiting base constructed in the non-archimedean collapse picture. Steinness conjecture. The…

Let ${X}^{an}$ be the smooth analytic space associated with the Calabi–Yau variety over $K={{\bf C}}((t))$, and let $\pi_{mer}$ map meromorphic points to their limiting points in t…

Let $(X,L)$ be a polarized manifold. Let $\mathcal{H}$ be the space of Kähler metrics in the class $c_1(L)$, let $\mathcal{H}^{\operatorname{NA}}$ be the space of smooth non-Archim…

Let $V$ have dimension $n$, let $\rho$ be an irreducible $GL_n({\cal O})$-component of $C^\infty(Gr_{n-i,n}(\mathbb F))$, and let ${\cal D}_i$, $P_i$, and…

Let $V$ have dimension $n$, let $Val_i^\infty(V)$ denote the degree-$i$ smooth translation-invariant valuations, and identify valuations represented by functions on the relevant Gr…

Mixed hard Lefschetz and Hodge–Riemann conjecture. $Val^\infty(V)$ satisfies the mixed hard Lefschetz theorem for the subset ${\cal K}$, and it satisfies the mixed Hodge–Riemann re…

Finite-energy non-Archimedean equality conjecture. Equality is obtained provided one includes finite-energy non-Archimedean metrics.

Let $X$ be the underlying projective variety, let $X_\mathrm{na}$ be its non-Archimedean analytification, and let $\mathcal{M}^1_\mathrm{na}$ be the space of finite-energy non-Arch…

Tropicalization conjecture. For every lattice $L\subset K^n$, there exists $\alpha_L\in\mathbb R_{>0}$ such that, for every $\alpha\geq\alpha_L$, $Q_{L,\alpha}$ is the survival fun…

Odaka's conjecture. The volume growth of geodesic balls in $X$ is bounded below by the dimension of the essential skeleton:

Let $X\to \mathbb{D}^*_t$ be a polarised meromorphic degeneration of $n$-dimensional Calabi–Yau manifolds, with relatively ample line bundle $L$, and let $X_K^{an}$ be the associat…

Let $X$ be a $K$-variety. Weak non-Archimedean Green–Griffiths–Lang–Vojta conjecture. The following are equivalent: … This is the weak form of the non-Archimedean Green–Griffiths–L…

Let $K$ be a complete discrete valued field with equal characteristic $0$, and let $\mathcal{X}$ be a uniruled variety over $K$. Say that a proper geometrically integral smooth $K$…

Let $K$ be a complete discrete valued field with equal characteristic $0$, and let $\mathcal{X}$ be a smooth proper integral variety of dimension $n$ over $K$. Write…

Let $\pi\colon\mathcal{X}\to\Delta^{\rm conv}(\mathbb{C})$ be an affine finite-type log-terminal faithfully flat morphism, where…

Kontsevich--Soibelman's conjecture. As the family maximally degenerates, the Ricci-flat Kähler metric spaces $X_t$ should converge in the Gromov--Hausdorff topology to the base of…

Let $X$ be a smooth, irreducible projective Berkovich space over a non-Archimedean field $k$ of characteristic $0$, let $\rho$ be a PL metric on $K_X$, and let…

Let $\mathcal{P}$ be a system, let $\lambda\in\mathbb{R}$, and let $q$ be a maximizer of the toric non-archimedean $\mu$-entropy functional $\bm{\check{\mu}}_{\mathrm{NA}}^\lambda$…

Let $P$ be a polytope, let $u_T^{\mathrm{can}}$ be the canonical family, and define the convex function … Let $\mathrm{DF}(q)$ denote the Donaldson–Futaki invariant, let…