Using $L^2$-methods, we prove a vanishing theorem for tame harmonic bundles over quasi-compact Kähler manifolds in a very general setting. As a special case, we give a completely new proof of the Kodaira-type vanishing theorems for Higgs bundles due to Arapura. To prove our vanishing theorem, we construct a fine resolution of the Dolbeault complex for tame harmonic bundles via the complex of sheaves of $L^2$-forms, and we establish the Hörmander $L^2$-estimate and solve $(\bar {\partial }_E+\theta )$-equations for Higgs bundles $(E,\theta )$.

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

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 the geometric Satake equivalence for étale metaplectic covers of reductive group schemes and extend the Langlands parametrization of V. Lafforgue to genuine cusp forms defined on their associated covering groups.

We describe the $J$-invariant of a semisimple algebraic group $G$ over a generic splitting field of a Tits algebra of $G$ in terms of the $J$-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the $J$-invariant of groups of type $\mathrm {D}_n$. In the case of type $\mathrm {D}_n$ we also provide explicit formulas for the first component and in some cases for the second component of the $J$-invariant.

We define $p$-adic $\mathrm {BPS}$ or $p\mathrm {BPS}$ invariants for moduli spaces $\operatorname {M}_{\beta,\chi }$ of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field $F$. Our definition relies on a canonical measure $\mu _{\rm can}$ on the $F$-analytic manifold associated to $\operatorname {M}_{\beta,\chi }$ and the $p\mathrm {BPS}$ invariants are integrals of natural ${\mathbb {G}}_m$ gerbes with respect to $\mu _{\rm can}$. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a $\chi$-independence result for these $p\mathrm {BPS}$ invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p\mathrm {BPS}$ with usual $\mathrm {BPS}$ invariants through a result of Maulik and Shen [Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586].

We prove that there exists at least one positive Einstein metric on $\mathbb {HP}^{m+1}\sharp \overline {\mathbb {HP}}^{m+1}$ for $m\geq ~2$. Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on $\mathbb {HP}^{m+1}\sharp \overline {\mathbb {HP}}^{m+1}$. We also investigate the existence of cohomogeneity-one positive Einstein metrics on $\mathbb {S}^{4m+4}$ and prove the existence of a non-standard Einstein metric on $\mathbb {S}^8$.

We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$, the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact growth rate $x^{1+2/{n(n-1)}}$, times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.

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.

Let $X$ and $Y$ be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length $n$ subschemes of a $K3$ surface. A class in $H^{p,p}(X\times Y,{\mathbb {Q}})$ is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let $f:H^2(X,{\mathbb {Q}})\rightarrow H^2(Y,{\mathbb {Q}})$ be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that $f$ is induced by an analytic correspondence. We furthermore lift $f$ to an analytic correspondence $\tilde {f}: H^*(X,{\mathbb {Q}})[2n]\rightarrow H^*(Y,{\mathbb {Q}})[2n]$, which is a Hodge isometry with respect to the Mukai pairings and which preserves the gradings up to sign. When $X$ and $Y$ are projective, the correspondences $f$ and $\tilde {f}$ are algebraic.

We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide with the homotopy groups of the associated Kan complex. We use this to prove a conjecture of Babson, Barcelo, de Longueville, and Laubenbacher from 2006, and a strong version of the Hurewicz theorem in discrete homotopy theory.

Compared with algebraic varieties the local monodromy of Drinfeld modules appears to be hopelessly complex: the image of the wild inertia subgroup under Tate module representations is infinite save for the case of potential good reduction. Nonetheless, we show that Tate modules of Drinfeld modules are ramified in a limited way: the image of a sufficiently deep ramification subgroup is trivial. This leads to a new invariant, the local conductor of a Drinfeld module. We establish an upper bound on the conductor in terms of the volume of the period lattice. As an intermediate step we develop a theory of normed lattices in function field arithmetic including the notion of volume. We relate normed lattices to vector bundles on projective curves. With the aid of Castelnuovo–Mumford regularity this implies a volume bound on norms of lattice generators, and the conductor inequality follows. Last but not least we describe the image of inertia for Drinfeld modules with period lattices of rank $1$. Just as in the theory of local $\ell$-adic Galois representations this image is commensurable with a commutative unipotent algebraic subgroup. However, in the case of Drinfeld modules such a subgroup can be a product of several copies of $\mathbf {G}_a$.

Inspired by K. Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety $\mathcal {X}$ of relative dimension $n$ is maximal when $\mathcal {X}$ is the projective space over the integers, endowed with the Fubini–Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when $n\leq 6$ (the extension to higher dimensions is conditioned on a conjectural ‘gap hypothesis’ for the degree). Translated into toric Kähler geometry, this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. Furthermore, we reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension $n$ it is shown how to control the height of the canonical toric model $\mathcal {X},$ with respect to the Kähler–Einstein metric, by the degree of $\mathcal {X}$. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.

We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.

We prove the Hasse principle for a smooth projective variety $X\subset \mathbb {P}^{n-1}_\mathbb {Q}$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $\mathbb {Q}$.

The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$-categories and diagrams of $(\infty, 1)$-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening–unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories.

The Haefliger–Thurston conjecture predicts that Haefliger's classifying space for $C^r$-foliations of codimension $n$ whose normal bundles are trivial is $2n$-connected. In this paper, we confirm this conjecture for piecewise linear (PL) foliations of codimension $2$. Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension $2$ and prove the simplicity of the identity component of PL surface homeomorphisms.

The $4 n^2$-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$, and obtain a $2 n^2$-inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb {P}^1$ satisfying the $K^2$-condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point.

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.