We introduce the notion of pseudo-Néron model and give new examples of varieties admitting pseudo-Néron models other than Abelian varieties. As an application of pseudo-Néron models, given a scheme admitting a finite morphism to an Abelian scheme over a positive-dimensional base, we prove that for a very general genus-0 degree-d curve in the base with d sufficiently large, every section of the scheme over the curve is contained in a unique section over the entire base.

In this paper, we associate with isometries of $\mathrm{CAT}(0)$ cube complexes specific subspaces, referred to as median sets, which play a similar role as minimizing sets of semisimple isometries in $\mathrm{CAT}(0)$ spaces. Various applications are deduced, including a cubulation of centralizers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a $\mathrm{CAT}(0)$ cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on $\mathrm{CAT}(0)$ cube complexes.

We extend the notion of Frobenius Betti numbers to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. To do so, we introduce new invariants, which we call Frobenius Euler characteristics. We prove uniform convergence and upper semicontinuity results for Frobenius Betti numbers and Euler characteristics. These invariants detect the singularities of a ring, extending two results from the local to the global setting.

In this paper, via the contraction mapping principle, we give a proof of a Bloch-type theorem for normalized harmonic Bochner–Takahashi K-mappings and for solutions to equations of the form $\mathit{P}\mathit{u}=0$, where P is a homogeneous differential operator with an analytic fundamental solution, that is, homogeneous elliptic operators with constant coefficients.

In this paper, we give complex geometric descriptions of the notions of algebraic geometric positivity of vector bundles and torsion-free coherent sheaves, such as nef, big, pseudo-effective, and weakly positive, by using singular hermitian metrics. As an application, we obtain a generalization of Mori’s result. We also give a characterization of the augmented base locus by using singular hermitian metrics on vector bundles and the Lelong numbers.

In this paper we prove that del Pezzo fibrations admit ${\mathbb{G}}_{\mathit{a}}^3$-structures if and only if they are ${\mathbb{P}}^2$-bundles over ${\mathbb{P}}^1$.

Let X be a simply connected rational elliptic space of formal dimension m, and let $\mathcal{E}(\mathit{X})$ denote the group of homotopy classes of self-equivalences of X. If Y is the space obtained by attaching rational cells of dimension q to X, where $\mathit{q}>2\mathit{m}+1$, then we prove that $\mathcal{E}(\mathit{Y})\cong \mathit{GL}(\mathit{r},\mathbb{Q})\times \mathcal{E}(\mathit{X})$ and ${\mathcal{E}}_{\ast }(\mathit{Y})\cong {\mathcal{E}}_{\ast }(\mathit{X})$, where $\mathit{r}=\dim {\mathit{H}}_{\mathit{q}}(\mathit{Y},\mathbb{Q})$. Here ${\mathcal{E}}_{\ast }(\mathit{X})$ denotes the subgroup of $\mathcal{E}(\mathit{X})$ of the elements inducing the identity on the homology groups. Consequently, we show that, for any finite group G and for any $\mathit{r}\in \mathbb{N}$, there exists a simply connected space X such that $\mathcal{E}(\mathit{X})\cong \mathit{GL}(\mathit{r},\mathbb{Q})\times \mathit{G}$.

We study behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of nonlocal type and is expressed as a convolution

$$\mathit{i}{\mathit{u}}_{\mathit{t}}+\mathrm{\Delta }\mathit{u}+(|\mathit{x}{|}^{-(\mathit{N}-\mathit{\gamma })}\ast |\mathit{u}{|}^{\mathit{p}})|\mathit{u}{|}^{\mathit{p}-2}\mathit{u}=0,\mathit{x}\in {\mathbb{R}}^{\mathit{N}},\mathit{t}\in \mathbb{R}.$$

Our main goal is to understand global behavior of solutions of this equation in various settings. In this work we make an initial attempt towards this goal and study ${\mathit{H}}^1$ (finite energy) solutions. We first investigate the ${\mathit{H}}^1$ local well-posedness and small data theory. We then, in the intercritical regime ($0<\mathit{s}<1$), classify the behavior of ${\mathit{H}}^1$ solutions under the mass-energy assumption $\mathcal{ME}[{\mathit{u}}_0]<1$, identifying the sharp threshold for global versus finite time solutions via the sharp constant of the corresponding convolution type Gagliardo–Nirenberg interpolation inequality (note that the uniqueness of a ground state is not known in the general case). In particular, depending on the size of the initial mass and gradient, solutions will either exist for all time and scatter in ${\mathit{H}}^1$, or blow up in finite time, or diverge along an infinite time sequence. To obtain ${\mathit{H}}^1$ scattering or divergence to infinity, in this paper we employ the well-known concentration compactness and rigidity method of Kenig and Merle [36] with the novelty of studying the nonlocal, convolution nonlinearity.