We prove the ascending chain condition for minimal log discrepancies on an arbitrary fixed threefold.
We construct relatively bounded toroidal and toric models of relatively bounded fibrations over curves.
In this paper, we give an explicit result on the Mordell conjecture over function field of characteristic zero. Based on the idea of Dimitrov–Gao–Habegger, we refine Vojta’s proof and use Looper–Silverman–Wilms’ explicit bound on small points to get our main theorem.
Assuming natural variational realization conjectures, we give uniform bounds for the obstruction to the integral Tate conjecture in 1-dimensional families of algebraic varieties over an infinite finitely generated field.
Assuming the Morrison-Kawamata cone conjecture for the generic fiber of a Calabi-Yau fibration and the abundance conjecture, we show (1) the finiteness of minimal models, (2) the existence of a weak rational polyhedral fundamental domain under the action of birational automorphism groups, and (3) the finiteness of varieties as targets of contractions. As an application, the finiteness of minimal models and the weak Morrison-Kawamata cone conjecture in relative dimensions $\leq 2$ are established.
