We systematically study the moduli stacks of Higgs bundles, spectral curves, and Norm maps on Deligne–Mumford curves. As an application, under some mild conditions, we prove the Strominger–Yau–Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.

We construct a collection of families of higher Chow cycles of type $(2,1)$ on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank $\ge 18$ in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.

The attractor conjecture for Calabi–Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. Using tools from transcendence theory, we provide a family of counterexamples to the attractor conjecture in almost all odd dimensions conditional on a specific case of the Zilber–Pink conjecture in unlikely intersection theory; these Calabi–Yau manifolds were first studied by Dolgachev. We also give constructions of new families of Calabi–Yau varieties, analogous to the mirror quintic family, with all middle Hodge numbers equal to one, which would also give counterexamples to the attractor conjecture.

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$, the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.

We prove that all smooth Fano threefolds in the families and are K-stable, and we also prove that smooth Fano threefolds in the family that satisfy one very explicit generality condition are K-stable.

We construct and study the moduli of stable hypersurfaces in toric orbifolds. Let X be a projective toric orbifold and $\alpha \in \operatorname{Cl}(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G = \operatorname{Aut}(X)$. Since the group G is non-reductive in general, we use new techniques of non-reductive geometric invariant theory. Using the A-discriminant of Gelfand, Kapranov and Zelevinsky, we prove semistability for quasismooth hypersurfaces of toric orbifolds. Further, we prove the existence of a quasi-projective moduli space of quasismooth hypersurfaces in a weighted projective space when the weighted projective space satisfies a certain condition. We also discuss how to proceed when this condition is not satisfied. We prove that the automorphism group of a quasismooth hypersurface of weighted projective space is finite excluding some low degrees.

For a real number $0<\epsilon <1/3$, we show that the anti-canonical volume of an $\epsilon $-klt Fano $3$-fold is at most $3,200/\epsilon ^4$, and the order $O(1/\epsilon ^4)$ is sharp.

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron–Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher-dimensional analogues of the Calabi–Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi–Yau pairs, and in each dimension at least three, there exist Schoen varieties with nonpolyhedral nef cone. We prove the Kawamata–Morrison–Totaro cone conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.

We prove an equality, predicted in the physical literature, between the Jeffrey–Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson–Thomas type invariants.

In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey–Kirwan residues are determined by the the Gross–Hacking–Keel mirror family to a log Calabi–Yau surface.

In this paper, we prove the nonvanishing and some special cases of the abundance for log canonical threefold pairs over an algebraically closed field k of characteristic $p> 3$. More precisely, we prove that if $(X,B)$ be a projective log canonical threefold pair over k and $K_{X}+B$ is pseudo-effective, then $\kappa (K_{X}+B)\geq 0$, and if $K_{X}+B$ is nef and $\kappa (K_{X}+B)\geq 1$, then $K_{X}+B$ is semi-ample.

As applications, we show that the log canonical rings of projective log canonical threefold pairs over k are finitely generated and the abundance holds when the nef dimension $n(K_{X}+B)\leq 2$ or when the Albanese map $a_{X}:X\to \mathrm {Alb}(X)$ is nontrivial. Moreover, we prove that the abundance for klt threefold pairs over k implies the abundance for log canonical threefold pairs over k.

For certain quasismooth Calabi–Yau hypersurfaces in weighted projective space, the Berglund-Hübsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror. We prove that the minimal log discrepancy of the quotient of such a hypersurface by its toric automorphism group is closely related to the weights and degree of the BHK mirror. As an application, we exhibit klt Calabi–Yau varieties with the smallest known minimal log discrepancy. We conjecture that these examples are optimal in every dimension.

In this article, we construct some examples of noncommutative projective Calabi–Yau schemes by using noncommutative Segre products and quantum weighted hypersurfaces. We also compare our constructions with commutative Calabi–Yau varieties and examples constructed in Kanazawa (2015, Journal of Pure and Applied Algebra 219, 2771–2780). In particular, we show that some of our constructions are essentially new examples of noncommutative projective Calabi–Yau schemes.

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture.

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of $K3^{[n]}$-type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of $0$-cycles on such varieties.

Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$. We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$-invariant, but lie in distinct path-connected components of the $\mu ^*$-constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$, respectively) make a Zariski pair of curves in $\mathbb {P}^2$, the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$-Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies.

We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field $\mathbf {k}$ and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when $\mathbf {k} = \mathbf {C}$. An application is given to the description of ‘canonical vector heights’ associated to such automorphism groups.

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$, these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$, we prove that they are different by exhibiting some walls.

We prove that the alpha invariant of a quasi-smooth Fano 3-fold weighted hypersurface of index $1$ is greater than or equal to $1/2$. Combining this with the result of Stibitz and Zhuang [SZ19] on a relation between birational superrigidity and K-stability, we prove the K-stability of a birationally superrigid quasi-smooth Fano 3-fold weighted hypersurfaces of index $1$.

We axiomatise the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalised automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery, we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.

The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${\mathcal {M}}^{\operatorname {GIT}}$, as a Baily–Borel compactification of a ball quotient ${(\mathcal {B}_4/\Gamma )^*}$, and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$. The spaces ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.