We define kappa classes on moduli spaces of Kollár-Shepherd-Barron-Alexeev (KSBA)-stable varieties and pairs, generalizing the Miller–Morita–Mumford classes on moduli of curves, and computing them in some cases where the virtual fundamental class is known to exist, including Burniat and Campedelli surfaces. For Campedelli surfaces, an intermediate step is finding the Chow (same as cohomology) ring of the GIT quotient $(\mathbb {P}^2)^7//SL(3)$.

For a smooth affine group scheme G over the ring of p-adic integers and a cocharacter $\mu $ of G, we develop the deformation theory for G-$\mu $-displays over the prismatic site of Bhatt–Scholze, and discuss how our deformation theory can be interpreted in terms of prismatic F-gauges introduced by Drinfeld and Bhatt–Lurie. As an application, we prove the local representability and the formal smoothness of integral local Shimura varieties with hyperspecial level structure. We also revisit and extend some classification results of p-divisible groups.

A proof of a proposition in the published paper was erroneous. We clarify the effects in this erratum.

We consider faithful actions of simple algebraic groups on self-dual irreducible modules and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the dimension of the variety. We prove that in all but a finite list of cases, there is a dense open subset where the stabilizer of any point is conjugate to a fixed subgroup, called the generic stabilizer. We use these results to determine whether there exists a dense orbit. This in turn lets us complete the answer to the problem of determining all pairs of maximal connected subgroups of a classical group with a dense double coset.

We prove Chai's conjecture on the additivity of the base change conductor of semiabelian varieties in the case of Jacobians of proper curves. This includes the first infinite family of non-trivial wildly ramified examples. Along the way, we extend Raynaud's construction of the Néron lft-model of a Jacobian in terms of the Picard functor to arbitrary seminormal curves (beyond which Jacobians admit no Néron lft-models, as shown by our more general structural results). Finally, we investigate the structure of Jacobians of (not necessarily geometrically reduced) proper curves over fields of degree of imperfection at most one and prove two conjectures about the existence of Néron models and Néron lft-models due to Bosch–Lütkebohmert–Raynaud for Jacobians of general proper curves in the case of perfect residue fields, thus strengthening the author's previous results in this situation.

Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably primitive permutation group is definably isomorphic to an algebraic permutation group living in the constants. Applications include the verification, in differentially closed fields, of the finite Morley rank permutation group conjectures of Borovik-Deloro and Borovik-Cherlin. Applying the results to binding groups for internality to the constants, it is deduced that if complete types p and q are of rank m and n, respectively, and are nonorthogonal, then the $(m+3)$rd Morley power of p is not weakly orthogonal to the $(n+3)$rd Morley power of q. An application to transcendence of generic solutions of pairs of algebraic differential equations is given.

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in characteristic $p\geq 7$. We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via G-Hilbert schemes. In the appendix, we discuss several approaches towards the notion of conjugacy classes for finite group schemes: This is an ingredient in McKay correspondences, but also of independent interest.

In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$-valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$-valued representations.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the ‘weakly adelic Mumford–Tate hypothesis’ and prove the generalised André–Pink–Zannier conjecture under this assumption, using Pila–Zannier strategy.

Let R be a discrete valuation ring of field of fractions K and of residue field k of characteristic $p> 0$. In an earlier work, we studied the question of extending torsors over K-curves into torsors over R-regular models of the curves in the case when the structural K-group scheme of the torsor admits a finite flat model over R. In this paper, we first give a simpler description of the problem in the case where the curve is semistable using recent work in Holmes, Molcho, Orecchia, and Poiret (2023, Journal für die Reine und Angewandte Mathematik [Crelle’s Journal] 230, 115–159) and Molcho and Wise (2022, Compositio Mathematica 158, 1477–1562). Second, if R is assumed to be Henselian and Japanese, we solve the problem of extending torsors by combining our previous work together with results in Antei and Emsalem (2018, Nagoya Mathematical Journal 230, 18–34) and Phung and Dos Santos (2023, Algebraic Geometry 230, 1–40), including the case where the structural group does not admit a finite flat R-model.

Let ${\mathcal {R}} \subset \mathbb {P}^1_{\mathbb {C}}$ be a finite subset of markings. Let G be an almost simple simply-connected algebraic group over $\mathbb {C}$. Let $K_G$ denote the compact real form of G. Suppose for each lasso l around the marked point, a conjugacy class $C_l$ in $K_G$ is prescribed. The aim of this paper is to give verifiable criteria for the existence of an irreducible homomorphism of $\pi _{1}(\mathbb P^1_{\mathbb {C}} \,{\backslash}\, {\mathcal {R}})$ into $K_G$ such that the image of l lies in $C_l$.

We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.

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.

We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools.

After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma $. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak {g}_{\Gamma }$ on category $ \mathcal {O}$ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.

To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

We prove that after inverting the Planck constant $h$, the Bezrukavnikov–Kaledin quantization $(X, {\mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {\mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.

We introduce an analogue of Bridgeland’s stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical Kähler metric, modelled on the constant scalar curvature Kähler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical Kähler metrics in the large volume regime. We also prove a local converse and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang–Mills equation.

Let ${\mathfrak g}$ be a complex simple Lie algebra and ${\mathfrak n}$ the nilradical of a parabolic subalgebra of ${\mathfrak g}$. We consider some properties of the coadjoint representation of ${\mathfrak n}$ and related algebras of invariants. This includes (i) the problem of existence of generic stabilizers, (ii) a description of the Frobenius semiradical of ${\mathfrak n}$ and the Poisson center of the symmetric algebra , (iii) the structure of as -module, and (iv) the description of square integrable (= quasi-reductive) nilradicals. Our main technical tools are the Kostant cascade in the set of positive roots of ${\mathfrak g}$ and the notion of optimization of ${\mathfrak n}$.

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.

Let G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$-grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$, let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$. We study the $G_0$-equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$. It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$, then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$-orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$, then both are $G_0$-prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$. Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$, then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$. As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of ${\mathfrak g}$ in place of ${\mathfrak g}_0$ or spherical nilpotent G-orbits in place of ${\mathcal O}_{\textsf{min}}$.