In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $-stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$, where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$, if $g(C)\leq 1$; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$.

We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_\infty $-coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures.

Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in ${\mathbb {P}}^8$ as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover $\operatorname {\mathrm {SU}}_C(2,L)$, the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2,8)$. In fact, each point $p\in C$ defines a natural embedding of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ in $G(2,8)$. We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ and thus deserves to be coined the Coble quadric of the pointed curve $(C,p)$.

We consider cyclic unramified coverings of degree d of irreducible complex smooth genus $2$ curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2, 3, 5, 7$ are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d\ge 11$ prime such that $\frac {d-1}2$ is also prime. We use results of arithmetic nature on $GL_2$-type abelian varieties combined with theta-duality techniques.

Divisorial stability of a polarised variety is a stronger – but conjecturally equivalent – variant of uniform K-stability introduced by Boucksom–Jonsson. Whereas uniform K-stability is defined in terms of test configurations, divisorial stability is defined in terms of convex combinations of divisorial valuations on the variety.

We consider the behaviour of divisorial stability under finite group actions and prove that equivariant divisorial stability of a polarised variety is equivalent to log divisorial stability of its quotient. We use this and an interpolation technique to give a general construction of equivariantly divisorially stable polarised varieties.

In this paper, we prove some orthogonality relations for representations arising from deep level Deligne–Lusztig schemes of Coxeter type. This generalizes previous results of Lusztig [Lus04], and of Chan and the second author [CI21b]. Applications include the study of smooth representations of p-adic groups in the cohomology of p-adic Deligne–Lusztig spaces and their relation to the local Langlands correspondences. Also, the geometry of deep level Deligne–Lusztig schemes gets accessible, in the spirit of Lusztig’s work [Lus76].

For $g\ge 2$ and $n\ge 0$, let $\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$ denote the complex moduli stack of n-marked smooth hyperelliptic curves of genus g. A normal crossings compactification of this space is provided by the theory of pointed admissible $\mathbb {Z}/2\mathbb {Z}$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of $\mathcal {H}_{g, n}$. Using this graph complex, we give a sum-over-graphs formula for the $S_n$-equivariant weight zero compactly supported Euler characteristic of $\mathcal {H}_{g, n}$. This formula allows for the computer-aided calculation, for each $g\le 7$, of the generating function $\mathsf {h}_g$ for these equivariant Euler characteristics for all n. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible G-covers of genus zero curves, when G is abelian, as a symmetric $\Delta $-complex. We use these complexes to generalize our formula for $\mathsf {h}_g$ to moduli spaces of n-pointed smooth abelian covers of genus zero curves.

Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $X_{\mathrm {reg}}$, and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.

We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F-regular type. As a consequence, all higher cohomologies of the theta line bundle vanish. During the proof, we develop a method to estimate codimension.

The Hankel index of a real variety X is an invariant that quantifies the difference between nonnegative quadrics and sums of squares on X. In [5], the authors proved an intriguing bound on the Hankel index in terms of the Green–Lazarsfeld index, which measures the ‘linearity’ of the minimal free resolution of the ideal of X. In all previously known cases, this bound was tight. We provide the first class of examples where the bound is not tight; in fact, the difference between Hankel index and Green–Lazarsfeld index can be arbitrarily large. Our examples are outer projections of rational normal curves, where we identify the center of projection with a binary form F. The Green–Lazarsfeld index of the projected curve is given by the complex Waring border rank of F [16]. We show that the Hankel index is given by the almost real rank of F, which is a new notion that comes from decomposing F as a sum of powers of almost real forms. Finally, we determine the range of possible and typical almost real ranks for binary forms.

Let $\pi $ be a cuspidal, cohomological automorphic representation of an inner form G of $\operatorname {{PGL}}_2$ over a number field F of arbitrary signature. Further, let $\mathfrak {p}$ be a prime of F such that G is split at $\mathfrak {p}$ and the local component $\pi _{\mathfrak {p}}$ of $\pi $ at $\mathfrak {p}$ is the Steinberg representation. Assuming that the representation is noncritical at $\mathfrak {p}$, we construct automorphic $\mathcal {L}$-invariants for the representation $\pi $. If the number field F is totally real, we show that these automorphic $\mathcal {L}$-invariants agree with the Fontaine–Mazur $\mathcal {L}$-invariant of the associated p-adic Galois representation. This generalizes a recent result of Spieß respectively Rosso and the first named author from the case of parallel weight $2$ to arbitrary cohomological weights.

An example of Cornalba and Shiffman from 1972 disproves in dimension two or higher a classical prediction that the count of zeros of holomorphic self-mappings of the complex linear space should be controlled by the maximum modulus function. We prove that such a bound holds for a modified coarse count inspired by the theory of persistence modules originating in topological data analysis.

Let $\Gamma $ be a finite group, let $\theta $ be an involution of $\Gamma $ and let $\rho $ be an irreducible complex representation of $\Gamma $. We bound ${\operatorname {dim}} \rho ^{\Gamma ^{\theta }}$ in terms of the smallest dimension of a faithful $\mathbb {F}_p$-representation of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$, where p is any odd prime and $\operatorname {\mathrm {Rad}}_p(\Gamma )$ is the maximal normal p-subgroup of $\Gamma $.

This implies, in particular, that if $\mathbf {G}$ is a group scheme over $\mathbb {Z}$ and $\theta $ is an involution of $\mathbf {G}$, then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf {G}(\mathbb {Z}_p)/ \mathbf {G} ^{\theta }(\mathbb {Z}_p) \right)$ is bounded, uniformly in p.

We study a quiver description of the nested Hilbert scheme of points on the affine plane and its higher rank generalization – that is, the moduli space of flags of framed torsion-free sheaves on the projective plane. We show that stable representations of the quiver provide an ADHM-like construction for such moduli spaces. We introduce a natural torus action and use equivariant localization to compute some of their (virtual) topological invariants, including the case of compact toric surfaces. We conjecture that the generating function of holomorphic Euler characteristics for rank one is given in terms of polynomials in the equivariant weights, which, for specific numerical types, coincide with (modified) Macdonald polynomials. From the physics viewpoint, the quivers we study describe a class of surface defects in four-dimensional supersymmetric gauge theories in terms of nested instantons.

Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given $\mu>0$, there exists $n_0$ such that the following holds. Every n-vertex Steiner triple system contains all hypertrees with at most $(1-\mu )n$ vertices whenever $n\geq n_0$. We prove this conjecture.

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a poly-logarithmic bound on an average, for a large class of reductive groups. The method is based on Arthur’s development of the spectral side of the trace formula, and ideas of Finis, Lapid and Müller.

As applications of our method, we prove the optimal lifting property for $\mathrm {SL}_n(\mathbb {Z}/q\mathbb {Z})$ for square-free q, as well as the Sarnak–Xue [52] counting property for the principal congruence subgroup of $\mathrm {SL}_n(\mathbb {Z})$ of square-free level. This makes the recent results of Assing–Blomer [8] unconditional.

Let $p \geq 5$ be a prime number, and let $G = {\mathrm {SL}}_2(\mathbb {Q}_p)$. Let $\Xi = {\mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let $\mathcal {R} \subset \Xi \times \Xi $ denote the support of the pro-p Iwahori ${\mathrm {Ext}}$-algebra of G, viewed as a $(Z,Z)$-bimodule. We show that the locally ringed space $\Xi /\mathcal {R}$ is a projective algebraic curve over ${\mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of $\Xi /\mathcal {R}$, we construct a stable localising subcategory $\mathcal {L}_U$ of the category of smooth k-linear representations of G.

We consider a dilute fully spin-polarized Fermi gas at positive temperature in dimensions $d\in \{1,2,3\}$. We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order $a^d\rho ^{2+2/d}$, where a is the p-wave scattering length of the repulsive interaction and $\rho $ is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260).

This paper is devoted to the global analysis of the three-dimensional axisymmetric Navier–Stokes–Maxwell equations. More precisely, we are able to prove that, for large values of the speed of light $c\in (c_0, \infty )$, for some threshold $c_0>0$ depending only on the initial data, the system in question admits a unique global solution. The ensuing bounds on the solutions are uniform with respect to the speed of light, which allows us to study the singular regime $c\rightarrow \infty $ and rigorously derive the limiting viscous magnetohydrodynamic (MHD) system in the axisymmetric setting.

The strategy of our proofs draws insight from recent results on the two-dimensional incompressible Euler–Maxwell system to exploit the dissipative–dispersive structure of Maxwell’s system in the axisymmetric setting. Furthermore, a detailed analysis of the asymptotic regime $c\to \infty $ allows us to derive a robust nonlinear energy estimate which holds uniformly in c. As a byproduct of such refined uniform estimates, we are able to describe the global strong convergence of solutions toward the MHD system.

This collection of results seemingly establishes the first available global well-posedness of three-dimensional viscous plasmas, where the electric and magnetic fields are governed by the complete Maxwell equations, for large initial data as $c\to \infty $.