We show that the double dual EPW sextic associated with a strongly smooth Gushel–Mukai surface can be realized as a moduli space of semistable objects on its bounded derived category. Also, we observe that the double dual EPW surface associated with a special Gushel–Mukai threefold can be realized as a moduli space of semistable objects on its Kuznetsov component. Then we discuss extensions of our main results to double EPW sextics and double EPW surfaces and a refinement of a statement of Bayer and Perry about Gushel–Mukai threefolds with equivalent Kuznetsov components, under a mild assumption.

We prove a conjecture of Manin and Orlov: the moduli space $\overline{\mathcal{M}}_{0,n}$ of stable rational curves with n marked points admits a full exceptional collection that is equivariant under the action of the symmetric group $S_n$ permuting the marked points. In particular, its $K_0$-group with integral coefficients is a permutation $S_n$-lattice.

A classical theorem of Siegel gives the average number of lattice points in bounded subsets of Euclidean space. Motivated by this, Veech introduced the Siegel–Veech formula for translation surfaces, describing the average number of saddle connections of bounded length. However, no such formula is known for flat surfaces with cone angles that are irrational multiples of $\pi$. A convex flat cone sphere is the Riemann sphere with a flat cone metric whose cone angles lie in $(0,2\pi)$. In this paper, we extend the Siegel–Veech theory to convex flat cone spheres by introducing generalized transforms that are uniformly bounded on moduli space and give rise to Siegel–Veech measures on the positive real line. We prove that these measures are absolutely continuous and piecewise real analytic. In contrast to translation surfaces, which benefit from $GL(2,\mathbb{R})$ dynamics, our approach instead uses tools from o-minimality. Finally, we analyze the asymptotic behavior of the Siegel–Veech measure near zero, obtaining results that parallel the computation of Siegel–Veech constants in the translation surface case.

Let $A$ be an abelian variety with totally degenerate reduction over a non-Archimedean field. We describe the moduli space of semi-homogeneous vector bundles on $A$ from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For $H=0$ our moduli space may be identified with the moduli space $M_{0,r}(A)$ of semi-stable vector bundles with vanishing Chern classes on $A$. In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of $A$ onto $M_{0,r}(A)$, which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of $M_{0,r}(A)$.

In this article, we investigate the p-rank stratification of the moduli space of curves of genus g that admit a double cover to a fixed elliptic curve E in characteristic $p>2$. We show that the closed p-rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of E of all the possible values of the p-rank on this moduli space.

We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this moduli space extends to a complete real hyperbolic orbifold structure on the space of stable real binary quintics. This turns the moduli space of stable real binary quintics into the quotient of the real hyperbolic plane by an explicit non-arithmetic triangle group.

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 study constant Q-curvature metrics conformal to the the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov–Hausdorff topology, has the local structure of a real algebraic variety with formal dimension equal to the number of the punctures. If a nondegeneracy hypothesis holds, we show that a neighbourhood in the moduli spaces is actually a smooth, real-analytic manifold of the expected dimension. We also construct a geometrically natural set of parameters, construct a symplectic structure on this parameter space and show that in the smooth case a small neighbourhood of the moduli space embeds as a Lagrangian submanifold in the parameter space. We remark that our construction of the symplectic structure is quite different from the one in the scalar curvature setting, due to the fact that the associated partial differential equation is fourth-order rather than second-order.

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.

A real variety whose real locus achieves the Smith–Thom equality is called maximal. This paper introduces new constructions of maximal real varieties, by using moduli spaces of geometric objects. We establish the maximality of the following real varieties:

• – moduli spaces of stable vector bundles of coprime rank and degree over a maximal real curve (recovering Brugallé–Schaffhauser’s theorem with a short new proof), which extends to moduli spaces of parabolic vector bundles;

• – moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal real curve, providing maximal hyper-Kähler manifolds in every even dimension;

• – if a real variety has maximal Hilbert square, then the variety and its Hilbert cube are maximal, which happens for all maximal real cubic 3-folds, but never for maximal real cubic 4-folds;

• – punctual Hilbert schemes on a maximal real surface with vanishing first $\mathbb {F}_2$-Betti number and connected real locus, such as $\mathbb {R}$-rational maximal real surfaces and some generalized Dolgachev surfaces;

• – moduli spaces of stable sheaves on an $\mathbb {R}$-rational maximal Poisson surface (e.g. the real projective plane).

We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group.

We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-Kähler manifold of $\mathrm {K3}^{[2]}$-type whose deformation space is smooth of dimension 10. Its moduli space is birational to a hyper-Kähler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-Kähler could lead to new examples of hyper-Kähler manifolds.

In this work, we study the Humbert-Edge curves of type 5, defined as a complete intersection of four diagonal quadrics in ${\mathbb{P}}^5$. We characterize them using Kummer surfaces, and using the geometry of these surfaces, we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we show how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-1}{2}$ and the moduli space of Humbert-Edge curves of type $n\geq 5$ where $n$ is an odd number.

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

We study the behaviour of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence.

By working in the ‘relative setting’ (splitting one node at a time), we demonstrate that there are many nonhereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show that this is true when irreducible components are replaced by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras.

In [12], Kim and the first author proved a result comparing the virtual fundamental classes of the moduli spaces of $\varepsilon $-stable quasimaps and $\varepsilon $-stable $LG$-quasimaps by studying localized Chern characters for $2$-periodic complexes.

In this paper, we study a K-theoretic analogue of the localized Chern character map and show that for a Koszul $2$-periodic complex it coincides with the cosection-localized Gysin map of Kiem and Li [11]. As an application, we compare the virtual structure sheaves of the moduli space of $\varepsilon $-stable quasimaps and $\varepsilon $-stable $LG$-quasimaps.