We are interested in the Klein–Gordon–Zakharov system in $\mathbb {R}^{1+2}$, which is an important model in plasma physics with extensive mathematical studies. The system can be regarded as semilinear coupled wave and Klein–Gordon equations with nonlinearities violating the null conditions. Without the compactness assumptions on the initial data, we aim to establish the existence of small global solutions, and in addition, we want to illustrate the optimal pointwise decay of the solutions. Furthermore, we show that the Klein–Gordon part of the system enjoys linear scattering, while the wave part has uniformly bounded low-order energy. None of these goals is easy because of the slow pointwise decay nature of the linear wave and Klein–Gordon components in $\mathbb {R}^{1+2}$. We tackle the difficulties by carefully exploiting the properties of the wave and the Klein–Gordon components, and by relying on the ghost weight energy estimates to close higher order energy estimates. This appears to be the first pointwise decay result and the first scattering result for the Klein–Gordon–Zakharov system in $\mathbb {R}^{1+2}$ without compactness assumptions.

In a previous paper (Farajzadeh-Tehrani in Geom Topol 26:989–1075, 2022), for any logarithmic symplectic pair (X, D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural Fredholm setup for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan–Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove that the subspace of simple maps in each stratum is cut transversely. Such perturbations enable a geometric construction of Gromov–Witten type invariants for certain semi-positive pairs (X, D) in arbitrary genera. In future works, we will use local perturbations and a gluing theorem to construct log Gromov–Witten invariants of arbitrary such pair (X, D).

For any pair of orientable closed hyperbolic 3-manifolds, this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski dense $\text {PSL}(2,{\mathbb {Q}}^{{\textrm{ac}}})$-representations of their fundamental groups, up to conjugacy; moreover, corresponding pairs of representations have identical invariant trace fields and isomorphic invariant quaternion algebras. (Here, ${\mathbb {Q}}^{{\textrm{ac}}}$ denotes an algebraic closure of ${\mathbb {Q}}$.) Next, assuming the p-adic Borel regulator injectivity conjecture for number fields, this paper shows that uniform lattices in $\text {PSL}(2,{\mathbb {C}})$ with isomorphic profinite completions have identical invariant trace fields, isomorphic invariant quaternion algebras, identical covolume, and identical arithmeticity.

In this note, we show that the solution of Kähler–Ricci flow on every Fano threefold from Family No. 2.23 in the Mori–Mukai’s list develops type II singularity. In fact, we show that no Fano threefold from Family No. 2.23 admits Kähler–Ricci soliton and the Gromov–Hausdorff limit of the Kähler–Ricci flow must be a singular $\mathbb {Q}$-Fano variety. This gives new examples of Fano manifolds of the lowest dimension on which Kähler–Ricci flow develops type II singularity.