Asian Journal of Mathematics

Volume 24 (2020)

Number 6

Divisorial instability and Vojta’s main conjecture for $\mathbb{Q}$-Fano varieties

Pages: 995 – 1006



Nathan Grieve (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)


We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround K‑stability for Fano varieties. There is an interpretation in terms of the barycentres of the Newton–Okounkov bodies. Our main results show how the property of K‑instability, combined with the valuation theoretic characterization thereof, which is made possible by work of K. Fujita and C. Li, implies instances of Vojta’s Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan’s Second Main Theorem that has been obtained by M. Ru and P. Vojta.


Vojta’s Main Conjecture, K-stability, Fano varieties, Diophantine approximation, Newton–Okounkov bodies

Received 17 September 2019

Accepted 13 February 2020

Published 3 September 2021