Volume 2023, Issue 801

Let a reductive group G act on a smooth variety X such that a good quotient X / / G {X/\!\!/G} exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of X / / G {X/\!\!/G} , obtained from a G -equivariant vector bundle on X , can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of X / / G {X/\!\!/G} . In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne–Mumford stacks.

Let E / Q \mathrm{E}/\mathbb{Q} be an elliptic curve and 𝑝 a prime of supersingular reduction for E \mathrm{E} . Consider a quadratic extension L / Q p L/\mathbb{Q}_{p} and the corresponding anticyclotomic Z p \mathbb{Z}_{p} -extension L ∞ / L L_{\infty}/L . We analyze the structure of the points E ⁢ ( L ∞ ) \mathrm{E}(L_{\infty}) and describe two global implications of our results.

We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p -adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p -adic Banach space representations of the group of units of a quaternion algebra over ℚ p {\mathbb{Q}_{p}} appearing in the p -adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases.

We show that the Jacobians of prestable curves over toroidal varieties always admit Néron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of such Jacobians. In particular, we show the existence of a maximal quasi-compact separated group model, which we call the saturated model, and has the extension property for all torsion sections. The Néron model and the saturated model coincide over a Dedekind base, so the saturated model gives an alternative generalization of the classical notion of Néron models to higher-dimensional bases; in the general case we give necessary and sufficient conditions for the Néron model and saturated model to coincide. The key result, from which most others descend, is that the logarithmic Jacobian of [S. Molcho and J. Wise, The logarithmic Picard group and its tropicalization, preprint 2018] is a log Néron model of the Jacobian.

In this paper, we study the hypercritical deformed Hermitian-Yang–Mills equation on compact Kähler manifolds and resolve two conjectures of Collins–Yau [Moment maps, nonlinear PDE, and stability in mirror symmetry, preprint (2018), https://arxiv.org/abs/1811.04824].

We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.

F.-H. Lin (1989) studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an obstacle to the higher regularity if the dimension is odd. We discuss the odd-dimensional case and study how the higher regularity is obstructed. We introduce the logarithmic distance to the boundary as an additional independent self-variable and establish concise boundary regularity for the minimal graph.

We provide several characterizations of collapsing and noncollapsing in convex ancient mean curvature flow, establishing in particular that collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane. As a consequence, we rule out collapsing singularity models in ( n - 1 ) {(n-1)} -convex mean curvature flow (even when the initial datum is only immersed). Explicit counterexamples show that ( n - 1 ) {(n-1)} -convexity is optimal. We are also able to rule out collapsing singularity models for suitably pinched solutions of higher codimension.