We obtain a topological and equivariant classification of closed, connected three-dimensional Alexandrov spaces admitting a local isometric circle action. We show, in particular, that such spaces are homeomorphic to connected sums of some closed $3$-manifold with a local circle action and finitely many copies of the suspension of the real projective plane.

This is a continuation of a series of articles studying the birational Mori fiber structures of anticanonically embedded $\mathbb{Q}$-Fano $3$-fold weighted complete intersections of codimension $2$. We have proved that $19$ families consist of birationally rigid varieties and $14$ families consist of birationally birigid varieties. Our aim here is to continue the work systematically and prove that, among the remaining $52$ families, $21$ families consist of birationally birigid varieties.

We show—for a semistable abelian variety $A_K$ over a complete discrete valuation field $K$—that every finite-subgroup scheme of $A_K$ extends to a log finite-flat group scheme over the valuation ring of $K$ endowed with the canonical log structure. To achieve this, we first give a positive answer to a question of Nakayama, namely whether every weak log-abelian variety over an fs (fine and saturated) log scheme with its underlying scheme locally noetherian is a sheaf for the Kummer-flat topology. We also give several equivalent conditions defining isogenies of log-abelian varieties.

In this article, we study the decay of the solutions of Schrödinger equations in the exterior of an obstacle. The main situations we are interested in are the general case (no nontrapping assumptions) and some weakly trapping situations.

On s’intéresse dans cet article à la décroissance temporelle de la solution du problème extérieur pour l’équation de Schrödinger, dans le cas d’obstacles. Les situations auquelles on s’intéresse sont le cas général (pas d’hypothèse géométrique) ou alors des situations faiblement captantes.

The Berglund–Hübsch–Krawitz (BHK) mirror symmetry construction applies to certain types of Calabi–Yau varieties that are birational to finite quotients of Fermat varieties. Their definition involves a matrix $A$ and a certain finite abelian group $G$, and we denote the corresponding Calabi–Yau variety by $Z_{A,G}$. The transpose matrix $A^T$ and the so-called dual group $G^T$ give rise to the BHK mirror variety $Z_{A^T,G^T}$. In the case of dimension 2, the surface $Z_{A,G}$ is a K3 surface of BHK type. Let $Z_{A,G}$ be a K3 surface of BHK type, with BHK mirror $Z_{A^T,G^T}$. Using work of Shioda, Kelly has shown that the geometric Picard number $\rho \left(Z_{A,G}\right)$ of $Z_{A,G}$ may be expressed in terms of a certain subset of the dual group $G^T$. We simplify this formula significantly to show that $\rho \left(Z_{A,G}\right)$ depends only upon the degree of the mirror polynomial $F_{A^T}$.

The only connected finite-dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, we determine existing adjunctions between its $1$-morphisms, and we find a minimal generating set with respect to the tensor structure. We also prove that, for the algebras mentioned above, every simple transitive $2$-representation of the $2$-category of projective bimodules is equivalent to a cell $2$-representation.

We define the notion of mixed Frobenius structure (MFS) which is a generalization of the structure of a Frobenius manifold. We construct an MFS on the cohomology of weak Fano toric surfaces and that of the three-dimensional projective space using local Gromov–Witten invariants. This is an analogue of the Frobenius manifold associated to the quantum cohomology in the local Calabi–Yau setting.

Let $\mathfrak{a}$ be an ideal of a commutative Noetherian (not necessarily local) ring $R$. In the case $cd(\mathfrak{a},R)\le 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is Abelian. Using this fact and the technique of way-out functors, we show that, if $cd(\mathfrak{a},R)\le 1$, or if $dim(R/\mathfrak{a})\le 1$, or if $\dim \left(R\right)\le 2$, then the local cohomology module $H_{\mathfrak{a}}^i\left(X\right)$ is $\mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i\in \mathbb{Z}$. We further answer Hartshorne’s Question 1.3 in the three aforementioned cases, and we reveal a correlation between his Questions 1.1, 1.2, and 1.3.

Nishino’s theorem asserts that a Stein family of Riemann surfaces is locally trivial if the fibers are $\mathbb{C}$. A new proof is given by applying a result on the extension of $L^2$ holomorphic functions.

Given an exceptional simple complex algebraic group $G$ and a symmetric pair $(G,K)$, we study the spherical nilpotent $K$-orbit closures in the isotropy representation of $K$. We show that they are all normal except in one case in type ${\mathsf{G}}_2$, and we compute the $K$-module structure of the ring of regular functions on their normalizations.

On démontre un analogue du théorème de Liouville effectif valable pour des points fermés sur une variété projective définie sur un corps de nombres. Ce résultat est une version effective d’un théorème récent de McKinnon et Roth. Une partie importante de cet article est consacrée à la démonstration d’une version effective d’un cas particulier d’un théorème remarquable de géométrie diophantienne dû à Faltings et Wüstholz. Combiné à de nouvelles comparaisons explicites entre l’évaluation d’une section d’un fibré en droites et une fonction distance donnée, ce résultat entraîne le théorème principal. Nous présentons également une autre approche, en montrant comment rendre effectifs les arguments de McKinnon et Roth. Ces deux points de vue conduisent à des versions distinctes du théorème principal, fournissant des majorations différentes de la hauteur des points satisfaisant une inégalité analogue à celle du théorème de Liouville classique.

We prove an effective analogue of Liouville’s theorem for closed points on an arbitrary projective variety defined over a number field. Our result can be interpreted as an effective version of a recent theorem proved by McKinnon and Roth. A central part of our work here is dedicated to giving an effective proof of a particular case of a powerful theorem in diophantine geometry proved by Faltings and Wüstholz. This result, combined with new explicit comparisons between evaluation of sections of a line bundle and a given distance function, leads to the expected theorem. We also deal with another approach, showing how to make the arguments of McKinnon and Roth effective. These two points of view lead to distinct versions of our main result, giving different upper bounds for the height of points satisfying a Liouville-type inequality.

We present and apply a theory of one-parameter $C_0$-semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of $C_0$-equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille–Yosida theorem. Then we particularize the theory in some functional spaces and identify two locally convex topologies that allow us to gather—under a unified framework—various notions of $C_0$-semigroups introduced by some authors to deal with Markov transition semigroups. Finally, we apply the results to transition semigroups associated to stochastic differential equations (SDEs).