Volume 2023, Issue 797

We study the arithmetic of degree N - 1 {N-1} Eisenstein cohomology classes for the locally symmetric spaces attached to GL N {\mathrm{GL}_{N}} over an imaginary quadratic field k . Under natural conditions we evaluate these classes on ( N - 1 ) {(N-1)} -cycles associated to degree N extensions L / k {L/k} as linear combinations of generalized Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of L -functions attached to Hecke characters of L as polynomials in Kronecker–Eisenstein series evaluated at torsion points on elliptic curves with complex multiplication by k . We recover in particular the algebraicity of these critical values.

We prove a Zariski–Nagata purity theorem for the motivic ramification filtration of a reciprocity sheaf. An important tool in the proof is a generalization of the Kato-Saito reciprocity map from geometric global class field theory to all reciprocity sheaves. As a corollary we obtain cut-by-curves and cut-by-surfaces criteria for various ramification filtrations. In some cases this reproves known theorems, in some cases we obtain new results.

We construct Kähler–Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler–Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler–Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler–Einstein metrics from hyperbolic geometry.

In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.

For an integer n ≥ 2 {n\geq 2} we define a polylogarithm ℒ ^ n {\widehat{\mathcal{L}}_{n}} , which is a holomorphic function on the universal abelian cover of ℂ ∖ { 0 , 1 } {\mathbb{C}\setminus\{0,1\}} defined modulo ( 2 ⁢ π ⁢ i ) n / ( n - 1 ) ! {(2\pi i)^{n}/(n-1)!} . We use the formal properties of its functional relations to define groups ℬ ^ k ⁢ ( F ^ ) {\widehat{\mathcal{B}}_{k}(\widehat{F})} lifting Goncharov’s Bloch groups ℬ k ⁢ ( F ) {\mathcal{B}_{k}(F)} of a field F , and show that they fit into a complex Γ ^ ⁢ ( F , n ) {\widehat{\Gamma}(F,n)} lifting Goncharov’s Bloch complex Γ ⁢ ( F , n ) {\Gamma(F,n)} . When F = ℂ {F=\mathbb{C}} , we show that the imaginary part (when n is even) or real part (when n is odd) of ℒ ^ n {\widehat{\mathcal{L}}_{n}} agrees with a real single valued polylogarithm ℒ n {\mathcal{L}_{n}} on the group H 1 ⁢ ( Γ ^ ⁢ ( ℂ , n ) ) {H^{1}(\widehat{\Gamma}(\mathbb{C},n))} . When n = 2 {n=2} , this group is Neumann’s extended Bloch group. Goncharov’s complex conjecturally computes the rational motivic cohomology of F , and one may speculate whether the extended complex computes the integral motivic cohomology. Finally, we use ℒ ^ 3 {\widehat{\mathcal{L}}_{3}} to construct a lift of Goncharov’s map H 5 ⁢ ( SL ⁡ ( 3 , ℂ ) ) → ℝ {H_{5}(\operatorname{SL}(3,\mathbb{C}))\to\mathbb{R}} to a complex valued map whose real part agrees with that of Goncharov. The lift makes use of the cluster ensemble structure on the Grassmannian Gr ⁡ ( 3 , 6 ) {\operatorname{Gr}(3,6)} .

The moduli space G g , d r → M g \mathcal{G}^{r}_{g,\smash{d}}\to\mathcal{M}_{g} parameterizing algebraic curves with a linear series of degree 𝑑 and rank 𝑟 has expected relative dimension ρ = g − ( r + 1 ) ⁢ ( g − d + r ) \rho=g-(r+1)(g-d+r) . Classical Brill–Noether theory concerns the case ρ ≥ 0 \rho\geq 0 ; we consider the non-surjective case ρ < 0 \rho<0 . We prove the existence of components of this moduli space with the expected relative dimension when 0 > ρ ≥ − g + 3 0>\rho\geq-g+3 , or 0 > ρ ≥ − C r ⁢ g + O ⁢ ( g 5 / 6 ) 0>\rho\geq-C_{r}g+\mathcal{O}(g^{5/6}) , where C r C_{r} is a constant depending on the rank of the linear series such that C r → 3 C_{r}\to 3 as r → ∞ r\to\infty . These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.

In this paper we study two functions F ⁢ ( x ) {F(x)} and J ⁢ ( x ) {J(x)} , originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark’s conjecture. We also discuss connections with 1-cocycles for the modular group PSL ⁡ ( 2 , ℤ ) {\operatorname{PSL}(2,\mathbb{Z})} .

We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kähler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kähler reduction, which is a hyperkähler reduction for a particular choice of the spectral function. The main tool for studying the system is a flat connection on the space of first-order deformations of the complex structure, that allows to obtain a formal complexification of the moment map equations. Using this connection, we describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions.

We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag–Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, A ~ 2 {\widetilde{A}_{2}} -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K -theory.