We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” ${\Bbb R}$-ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.

We apply Angehrn-Siu-Helmke’s method to estimate basepoint freeness thresholds of higher dimensional polarized abelian varieties. We showed that a conjecture of Caucci holds for very general polarized abelian varieties in the moduli spaces $\mathcal {A}_{g, l}$ with only finitely many possible exceptions of primitive polarization types l in each dimension g. We improved the bound of basepoint freeness thresholds of any polarized abelian $4$ -folds and simple abelian $5$ -folds.

We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .

Let $\pi \colon \mathcal {X}\to B$ be a family whose general fibre $X_b$ is a $(d_1,\,\ldots,\,d_a)$-polarization on a general abelian variety, where $1\leq d_i\leq 2$, $i=1,\,\ldots,\,a$ and $a\geq 4$. We show that the fibres are in the same birational class if all the $(m,\,0)$-forms on $X_b$ are liftable to $(m,\,0)$-forms on $\mathcal {X}$, where $m=1$ and $m=a-1$. Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class.

Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$, with the property that there are infinitely many hypersurfaces on $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$. In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$-varieties and of Cantat’s theorem to self-correspondences.

In this note, using methods introduced by Hacon et al. [‘Boundedness of varieties of log general type’, Proceedings of Symposia in Pure Mathematics, Volume 97 (American Mathematical Society, Providence, RI, 2018) 309–348], we study the accumulation points of volumes of varieties of log general type. First, we show that if the set of boundary coefficients Λ satisfies the descending chain condition (DCC), is closed under limits and contains 1, then the corresponding set of volumes satisfies the DCC and is closed under limits. Then, we consider the case of ε-log canonical varieties, for 0 < ε < 1. In this situation, we prove that if Λ is finite, then the corresponding set of volumes is discrete.

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper, we study affine-related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$, then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

We study a notion of ‘b-stability’, introduced previously by the author in connection with the existence of constant scalar curvature Kähler, and Kähler-Einstein, metrics. The main result is Theorem 1.2, which makes progress towards a statement that the existence of such metrics implies b-stability. The proof is a modification of an argument of Stoppa, taking account of the birational transformations involved in the definition of b-stability.

We establish here several “invariance of plurigenera type” theorems for twisted pluricanonical forms and metrics of adjoint ℝ-bundles.

We construct natural equivalences between derived categories of coherent sheaves on the local models for stratified Mukai and Atiyah flops (of type A).

In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{r}},\,d \right)$ , stabilize when $r\,\ge \,d$. We give a complete characterization of the effective divisors on ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{d}},\,d \right)$ . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

The moduli space of principally polarised abelian 4-folds can be compactified in several different ways by toroidal methods. Here we consider in detail the Igusa compactification and the (second) Voronoi compactification. We describe in both cases the cone of nef Cartier divisors. The proof depends on a detailed description of the Voronoi compactification, which makes it possible to proceed by induction, using the known description of the nef cone for compactifications of ${\mathcal A}_3$. The Igusa compactification has a non-${\mathbb Q}$-factorial singularity, which is resolved by a single blow-up: this resolution is the Voronoi compactification. The exceptional divisor $E$ is a toric Fano variety (of dimension 9): the other boundary divisor, $D$, corresponds to degenerations with corank~1. After imposing a level structure in order to avoid certain technical complications, we show that the closure of $D$ in the Voronoi compactification maps to the Voronoi compactification of ${\mathcal A}_3$. The toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed. This inductive process is only valid for the Voronoi compactification: the result for the Igusa compactification is deduced from the Voronoi compactification.