In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

We show that the anti-canonical volume of an $n$-dimensional Kähler–Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

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 smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,\mathfrak{m})$ be a local ring; we prove that if $R_{\text{red}}$ is Du Bois, then $H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ is surjective for every $i$. We find many applications of this result. For example, we answer a question of Kovács and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969–1000; MR 3531359] on the Cohen–Macaulay property of Du Bois singularities. We obtain results on the injectivity of $\operatorname{Ext}$ that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600] in characteristic $0$. These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen–Macaulayness of the defining ideal of Du Bois singularities, which are characteristic-$0$ analogs and generalizations of results of Singh–Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147–155]. We extend results on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172; MR 0417172 (54 #5230)]. We also prove that singularities of dense $F$-injective type deform.

We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of Kawamata log terminal (klt) singularities. As an application, we verify that the local fundamental group of a three-dimensional klt singularity and the fundamental group of the smooth locus of a three-dimensional Fano variety with canonical singularities are always finite.

The author finds a limit on the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. This result is then used in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain nonnormal del Pezzo surfaces over bases of small dimension.

A singularity in characteristic zero is said to be of dense $F$-pure type if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. We prove that if $x\in X$ is an isolated log canonical singularity with $\mu (x\in X)\leq 2$ (where the invariant $\mu $ is as defined in Definition 1.4), then it is of dense $F$-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense $F$-pure type in the case of three-dimensional isolated $ \mathbb{Q} $-Gorenstein normal singularities.

In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being ℚ-Gorenstein or the pair being log ℚ-Gorenstein. The main features of the theory extend to this setting in a natural way.

The paper gives fundamental results on the universal abelian covers of rational surface singularities. Let $(X,o)$ be a normal complex surface singularity germ with a rational homology sphere link. Then $(X,o)$ has the universal abelian cover $(Y,o) \longrightarrow (X,o)$. It is shown that if $(X,o)$ is rational or minimally elliptic, and if it has a star-shaped resolution graph, then $(Y,o)$ is a complete intersection (a partial answer to the conjecture of Neumann and Wahl). A way is given to compute the multiplicity and the embedding dimension of $(Y,o)$ from the resolution graph of $(X,o)$ in the case when $(X,o)$ is rational.

We give a geometric description of the loci in the arc space defined by order of contact with a given subscheme, using the resolution of singularities. This induces an identification of the valuations defined by cylinders in the arc space with divisorial valuations. In particular, we recover the description of invariants coming from the resolution of singularities in terms of arcs and jets.