The moduli space of bundle stable pairs $\overline {M}_C(2,\Lambda )$ on a smooth projective curve C, introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline {M}_C(2,\Lambda )$ to that of the moduli spaces of stable vector bundles $\overline {N}_C(2,\Lambda )$. In particular, Thaddeus’ moduli spaces for genus two curves are all K-stable.

Let X be a smooth proper rigid analytic space over a complete algebraically closed field extension K of $\mathbb {Q}_p$. We establish a Hodge–Tate decomposition for X with G-coefficients, where G is any commutative locally p-divisible rigid group. This generalizes the Hodge–Tate decomposition of Faltings and Scholze, which is the case $G=\mathbb {G}_a$. For this, we introduce geometric analogs of the Hodge–Tate spectral sequence with general locally p-divisible coefficients. We prove that these spectral sequences degenerate at $E_2$. Our results apply more generally to a class of smooth families of commutative adic groups over X and in the relative setting of smooth proper morphisms $X\rightarrow S$ of seminormal rigid spaces. We deduce applications to analytic Brauer groups and the geometric p-adic Simpson correspondence.

Using metric techniques introduced by Berndtsson, we show a result on the constancy of families dominated by a constant variety and, on the opposite side, results on the strong non-isotriviality of certain families of surfaces with positive index and, in arbitrary dimension, in terms of the complex conjugate of a suitable representative of the Kodaira–Spencer class. We also give a metric interpretation of the liftability of relative volume forms.

In this article, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For a threefold, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which either maps to a curve or is fibered by curves. In practice, we treat arbitrary dimensional irregular varieties with either one-dimensional Albanese fiber or one-dimensional Albanese image. We prove that such a variety carries another fibration transversal to its Albanese morphism (a “bi-fibration” structure), which is an analog structure of bielliptic or quasi-bielliptic surfaces. In turn, we give an explicit description of irregular threefolds with trivial canonical divisors.

We present an explicit geometric invariant theory construction which produces both the minimal resolution of the type $D_4$ surface singularity, and also the orbifold resolution. Our construction is based on a Tannakian approach which is in principle applicable to larger groups.

We construct moduli stacks of stable sheaves for surfaces fibered over marked nodal curves by using expanded degenerations. These moduli stacks carry a virtual class and therefore give rise to enumerative invariants. In the case of a surface with two irreducible components glued along a smooth divisor, we prove a degeneration formula that relates the moduli space associated to the surface with the relative spaces associated to the two components. For a smooth surface and no markings, our notion of stability agrees with slope stability with respect to a suitable choice of polarization. We apply our results to compute elliptic genera of moduli spaces of stable sheaves on some elliptic surfaces.

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers.

For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let $\pi :D \to \Gamma \backslash D = S$ realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of $\Gamma < G$, an arithmetic subgroup. Let $S' \subseteq S$ be a special subvariety of S realized as $\pi (D')$ for $D' \subseteq D$ a homogeneous space for an algebraic subgroup of G. Let $X \subseteq S$ be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with $\pi (gD')$ is persistently likely as g ranges through G with $\pi (gD')$ a special subvariety of S, meaning that whenever $\zeta :S_1 \to S$ and $\xi :S_1 \to S_2$ are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with $\zeta $ finite, $\dim \xi \zeta ^{-1} X + \dim \xi \zeta ^{-1} \pi (gD') \geq \dim \xi S_1$. Then $X \cap \bigcup _{g \in G, \pi (g D') \text { is special }} \pi (g D')$ is dense in X for the Euclidean topology.

We compute the integral Chow rings of $\overline {\mathcal {M}}_{1,n}$ for $n=3,4$. The alternative compactifications introduced by Smyth – and studied further by Lekili and Polishchuk – present each of these stacks as a sequence of weighted blow-ups and blow-downs from a weighted projective space. We compute all the integral Chow rings by repeated application of the blow-up formula.

We define and study a generalization of the Beilinson–Drinfeld Grassmannian to the case where the curve is replaced by a smooth projective surface $X$, and the trivialization data are given on loci associated with a nonlinear flag of closed subschemes. We first establish some general formal gluing results for moduli of (almost) perfect complexes and torsors. We construct a simplicial object $\operatorname {\underline {\mathsf{Fl}}}_X$ of flags of closed subschemes of a smooth projective surface $X$, associated with the operation of taking union of flags. We prove that this simplicial object has the $2$-Segal property. For an affine complex algebraic group $G$, we define a derived, flag analogue $\mathcal{G}r_X$ of the Beilinson–Drinfeld Grassmannian of $G$-bundles on the surface $X$, and show that most of the properties of the Beilinson–Drinfeld Grassmannian for curves can be extended to our flag generalization: we prove a factorization formula, the existence of a canonical flat connection and define a chiral product on suitable sheaves on $\operatorname {\underline {\mathsf{Fl}}}_X$ and on $\mathcal{G}r_X$. We sketch the construction of actions of flags analogues of the loop group and of the positive loop group on $\mathcal{G}r_X$. To fixed ‘large’ flags on $X$, we associate ‘exotic’ derived structures on the stack of $G$-bundles on $X$.

We construct moduli spaces of objects in an abelian category satisfying some finiteness hypotheses. Our approach is based on the work of Artin and Zhang [Algebr. Represent. Theory 4 (2001), 305–394] and the intrinsic construction of moduli spaces for stacks developed by Alper, Halpern-Leistner and Heinloth [Invent. Math. 234 (2023), 949–1038].

Using the $\infty $-categorical enhancement of mixed Hodge modules constructed by the author in a previous paper, we explain how mixed Hodge modules canonically extend to algebraic stacks, together with all the six operations and weights. We also prove that Drew’s approach to motivic Hodge modules gives an $\infty $-category that embeds fully faithfully in mixed Hodge modules, and we identify the image as mixed Hodge modules of geometric origin.

We give an interpretation of the semi-infinite intersection cohomology sheaf associated with a semisimple simply connected algebraic group in terms of finite-dimensional geometry. Specifically, we describe a procedure for building factorization spaces over moduli spaces of finite subsets of a curve from factorization spaces over moduli spaces of divisors, and show that, under this procedure, the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology (IC) sheaf in the factorizable Grassmannian. We define ‘semi-infinite t-structures’ for a large class of schemes with an action of the multiplicative group, and show that, for the Zastava, the limit of these t-structures recovers the infinite-dimensional version. As an application, we also construct factorizable parabolic semi-infinite IC sheaves and a generalization (of the principal case) to Kac–Moody algebras.

We prove that the non-properness set of a local homeomorphism $\mathbb R^n \to \mathbb R^n$ cannot be ambient homeomorphic to an affine subspace of dimension n − 2. This particularly provides a partial positive answer to a conjecture of Jelonek, that claims the global invertibility of polynomial local diffeomorphisms having non-properness set with codimension greater than 1. Our reasons to obtain this result lead us to some properties of the non-properness set when it has codimension 2, the heart of Jelonek’s conjecture. We also provide a global injectivity theorem related to this conjecture that turns out to generalize previous results of the literature.

Motivated by classical Alexander invariants of affine hypersurface complements, we endow certain finite dimensional quotients of the homology of abelian covers of complex algebraic varieties with a canonical and functorial mixed Hodge structure (MHS). More precisely, we focus on covers which arise algebraically in the following way: if U is a smooth connected complex algebraic variety and G is a complex semiabelian variety, the pullback of the exponential map by an algebraic morphism $f:U\to G$ yields a covering space $\pi :U^f\to U$ whose group of deck transformations is $\pi _1(G)$. The new MHSs are compatible with Deligne’s MHS on the homology of U through the covering map $\pi $ and satisfy a direct sum decomposition as MHSs into generalized eigenspaces by the action of deck transformations. This provides a vast generalization of the previous results regarding univariable Alexander modules by Geske, Maxim, Wang and the authors in [16, 17]. Lastly, we reduce the problem of whether the first Betti number of the Milnor fiber of a central hyperplane arrangement complement is combinatorial to a question about the Hodge filtration of certain MHSs defined in this paper, providing evidence that the new structures contain interesting information.

We study the moduli space of constant scalar curvature Kähler (cscK) surfaces around toric surfaces. To this end, we introduce the class of foldable surfaces: smooth toric surfaces whose lattice automorphism group contains a non-trivial cyclic subgroup. We classify such surfaces and show that they all admit a cscK metric. We then study the moduli space of polarised cscK surfaces around a point given by a foldable surface, and show that it is locally modelled on a finite quotient of a toric affine variety with terminal singularities.

We prove motivic versions of mass formulas by Krasner, Serre, and Bhargava concerning (weighted) counts of extensions of local fields.

We study the moduli stacks of slope-semistable torsion-free coherent sheaves that admit reflexive, respectively, locally free, Seshadri graduations on a smooth projective variety. We show that they are open in the stack of coherent sheaves and that they admit good moduli spaces when the field characteristic is zero. In addition, in the locally free case we prove that the resulting moduli space is a quasi-projective scheme.

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov–Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet–Tevelev.

We use a graph to define a new stability condition for algebraic moduli spaces of rational curves. We characterise when the tropical compactification of the moduli space agrees with the theory of geometric tropicalisation. The characterisation statement occurs only when the graph is complete multipartite.

Let $X$ be a very general Gushel–Mukai (GM) variety of dimension $n\geq 4$, and let $Y$ be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of $X$ and $Y$. In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an eight-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.