Volume 2019, Issue 746

Following Jacquet, Lapid and Rogawski, we define regularized periods of automorphic forms on Un+1\mathrm{U}_{n+1}×\timesUn\mathrm{U}_{n} along the diagonal subgroup Un{\mathrm{U}_{n}} and compute the regularized periods of cuspidal Eisenstein series and their residues. The formula for the periods of residues has an application to the Gan–Gross–Prasad conjecture.

We investigate weighted Sobolev spaces on metric measure spaces (X,d,𝔪){(X,\mathrm{d},\mathfrak{m})}. Denoting by ρ the weight function, we compare the space W1,p⁢(X,d,ρ⁢𝔪){W^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} (which always coincides with the closure H1,p⁢(X,d,ρ⁢𝔪){H^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} of Lipschitz functions) with the weighted Sobolev spaces Wρ1,p⁢(X,d,𝔪){W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})} and Hρ1,p⁢(X,d,𝔪){H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})} defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W1,p⁢(X,d,ρ⁢𝔪)=Hρ1,p⁢(X,d,𝔪){W^{1,p}(X,\mathrm{d},\rho\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{% m})}. We also adapt the results in [23] and in the recent paper [27] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality Wρ1,p⁢(X,d,𝔪)=Hρ1,p⁢(X,d,𝔪){W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},% \mathfrak{m})}.

We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold.

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension << 1 are free. On the other hand we construct for any ε >> 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension << 1 ++ ε.

There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in SU⁢(2){{\rm{SU}}(2)}. The main result in this paper shows that for any non-cyclic finite subgroup Γ ⊂\subset U(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of ℂ2\mathbb{C}^{2}/Γ, for which Γ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat Kähler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kähler metrics admitting holomorphic isometric circle actions in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.

In this article we prove that a connected and properly embedded translating soliton in ℝ3{{\mathbb{R}^{3}}} with uniformly bounded genus on compact sets which is C1{C^{1}}-asymptotic to two planes outside a cylinder, either is flat or coincide with the grim reaper cylinder.

We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed to give applications of this complete description. In this setting, we verify a question posed by Kawamata: we show that D -equivalence and K -equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne–Mumford stacks. This recovers Kawamata’s theorem that all projective toric Deligne–Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover and extend Orlov’s σ-model/Landau–Ginzburg model correspondence.

We supply an argument that was missing from the proof of the main result of the article “Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K -theory” [7]. The argument is essentially formal, and does not affect the strategy of the proof.