This book gives a comprehensive treatment of the singularities that appear in the minimal
model program and in the moduli problem for varieties. The study of these singularities and …

Let X be a normal complex projective variety with at worst klt singularities, and L a big line
bundle on X. We use valuations to study the log canonical threshold of L, as well as another …

We study asymptotic jumping numbers for graded sequences of ideals, and show that every
such invariant is computed by a suitable real valuation of the function field. We conjecture …

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He
conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with …

We use the $ D $-module theoretic description of generalized test ideals to show that in any
$ F $-finite regular ring the $ F $-thresholds of hypersurfaces are discrete and rational …

Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded
dimension satisfies the ascending chain condition. In this article we prove that the conjecture …

We establish the relative minimal model program with scaling for projective morphisms of
quasi-excellent algebraic spaces admitting dualizing complexes, quasi-excellent formal …

A conic bundle is a contraction between normal varieties of relative dimension such that is
relatively ample. We prove a conjecture of Shokurov that predicts that if is a conic bundle …

We generalize the rationality theorem of the accumulation points of log canonical thresholds,
which was proved by Hacon, McKernan, and Xu. Further, we apply the rationality to the ACC …

We prove a uniform bound on the growth of symbolic powers of arbitrary (not necessarily
radical) ideals in arbitrary (not necessarily excellent) regular rings of all characteristics. This …