Volume 2021, Issue 778

Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K -analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.

In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on BunG{\operatorname{Bun}_{G}} (that is, the difference between !{!}-extensions and *{*}-extensions) is controlled, Langlands-dually, by the locus of semisimple Gˇ{{\check{G}}}-local systems. To see this, we first rephrase the question in terms of Deligne–Lusztig duality and then study the Deligne–Lusztig functor 𝖣𝖫Gspec{\mathsf{DL}_{G}^{\mathrm{spec}}} acting on the spectral Langlands DG category IndCoh𝒩⁢(LSG){{\mathrm{IndCoh}}_{\mathcal{N}}({\mathrm{LS}}_{G})}. We prove that 𝖣𝖫Gspec{\mathsf{DL}_{G}^{\mathrm{spec}}} is the projection IndCoh𝒩⁢(LSG)↠QCoh⁢(LSG){{\mathrm{IndCoh}}_{\mathcal{N}}({\mathrm{LS}}_{G})\twoheadrightarrow{\mathrm{% QCoh}}({\mathrm{LS}}_{G})}, followed by the action of a coherent D -module StG∈𝔇⁢(LSG){{\mathrm{St}}_{G}\in\mathfrak{D}({\mathrm{LS}}_{G})}, which we call the Steinberg D -module. We argue that StG{{\mathrm{St}}_{G}} might be regarded as the dualizing sheaf of the locus of semisimple G -local systems. We also show that 𝖣𝖫Gspec{\mathsf{DL}_{G}^{\mathrm{spec}}}, while far from being conservative, is fully faithful on the subcategory of compact objects.

We construct an explicit, multiplicative Chow–Künneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga–Lunts–Verbitsky Lie algebra.

We investigate when a meromorphic connection on a smooth rigid analytic variety 𝑋 gives rise to a coadmissible D⏜X\overparen{\mathcal{D}}_{X}-module, and show that this is always the case when the roots of the corresponding 𝑏-functions are all of positive type. We also use this theory to give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.

We prove the other half of Brauer’s Height Zero Conjecture in the case of principal blocks.

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton ℝ4{\mathbb{R}^{4}}, or 𝕊3×ℝ{\mathbb{S}^{3}\times\mathbb{R}}, or 𝕊2×ℝ2.{\mathbb{S}^{2}\times\mathbb{R}^{2}.} In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

Let Ω be a proper open cone in a real Banach space V . We show that the tube domain V⊕i⁢Ω{V\oplus i\Omega} over Ω is biholomorphic to a bounded symmetric domain if and only if Ω is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that V is a unital JB-algebra in an equivalent norm and Ω is the interior of {v2:v∈V}{\{v^{2}:v\in V\}}.

Using log canonical thresholds and basis divisors Fujita–Odaka introduced purely algebro-geometric invariants δm{\delta_{m}} whose limit in m is now known to characterize uniform K-stability on a Fano variety. As shown by Blum–Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these δm{\delta_{m}}-invariants characterizes uniform Ding stability. A basic question since Fujita–Odaka’s work has been to find an analytic interpretation of these invariants. We show that each δm{\delta_{m}} is the coercivity threshold of a quantized Ding functional on the m th Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for ℙn{{\mathbb{P}}^{n}}. Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler–Ricci solitons (and the more general g -solitons of Berman–Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.