We prove that the coarse assembly maps for proper metric spaces that are nonpositively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces have bounded coarse geometry. Also it is shown that we can calculate the coarse $K$-homology and the $K$-theory of the Roe algebra by using the visual boundaries.

We show that the height of a variety over a finitely generated field of characteristic zero can be written as an integral of local heights over the set of places of the field. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to compute some arithmetic intersection numbers of nontoric arithmetic varieties over the rational numbers.

Let $(S,L)$ be a polarized abelian surface of Picard rank $1$, and let $\varphi$ be the function which takes each ample line bundle $L\text{'}$ to the least integer $k$ such that $L\text{'}$ is $k$-very ample but not $(k+1)$-very ample. We use Bridgeland’s stability conditions and Fourier–Mukai techniques to give a closed formula for $\varphi \left(L^n\right)$ as a function of $n$, showing that it is linear in $n$ for $n>1$. As a by-product, we calculate the walls in the Bridgeland stability space for certain Chern characters.

This is the second part of our three-part study titled “A few examples of local rings.” In the first part, we collected the basic tools and well-known important examples. But, since almost all special formal fibers of the examples there have extremely good properties, their derived normal rings are always Noetherian and analytically unramified. In this paper, we develop the basic ideas of the first part and get a somewhat improved method, which gives factorial local domains whose completion can be almost any complete local rings.

We establish two principles which state that, whenever an operator is bounded on a given Banach function space, then under some simple conditions, it is also bounded on the corresponding Morrey spaces and block spaces. By applying these principles on some concrete operators, we generalize the Fefferman–Stein vector-valued inequalities, define and study the Triebel–Lizorkin block spaces with variable exponents, and extend the mapping properties of the fractional integral operators to Morrey-type spaces and block-type spaces.

In this article, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this article asserts that any Frobenius-preserving isomorphism between the geometrically pro-$l$ fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-$l$ fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.

Let $T$ be a maximal torus of $PSL(n,\mathbb{C})$. For $n\ge 4$, we construct a smooth compactification of $PSL(n,\mathbb{C})/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{PSL(n,\mathbb{C})}$ for a suitable choice of $T$-linearized ample line bundle on $\overline{PSL(n,\mathbb{C})}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of $PSL(n,\mathbb{C})/T$ is $PSL(n,\mathbb{C})$ itself.

Let $\overline{G}=(G,\omega )$ be a vertex-weighted graph, and let $\delta$ be a divisor class on $G$. Let $r_{\overline{G}}\left(\delta \right)$ denote the (combinatorial) rank of $\delta$. Caporaso has introduced the algebraic rank $r_{\overline{G}}^{\mathrm{alg}}\left(\delta \right)$ of $\delta$ by using nodal curves with dual graph $\overline{G}$. In this paper, when $\overline{G}$ is hyperelliptic or of genus $3$, we show that $r_{\overline{G}}^{\mathrm{alg}}\left(\delta \right)\ge r_{\overline{G}}\left(\delta \right)$ holds, generalizing our previous result. We also show that, with respect to the specialization map from a nonhyperelliptic curve of genus $3$ to its reduction graph, any divisor on the graph lifts to a divisor on the curve of the same rank.