Let $p$ be an odd prime number, and $N$ a positive integer prime to $p$. We prove that $\mu$-type subgroups of the modular Jacobian variety $J_1(N)$ or $J_1(Np)$ of order a power of $p$ and defined over some abelian extensions of $\mathbb{Q}$ are trivial, under several hypotheses. For the proof, we use the method of Vatsal. As application, we show that a conjecture of Sharifi is valid in some cases.

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X}, KQ) = 0$ then $\mathcal{X}$ is projective. In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{\omega_1}$ such that, when $K$ is a countable field, $\mathbf{MA}_{\aleph_1}$ (Martin's axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1}, KQ) = 0$.

In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ is H-stable when $n \leq 2$ and there exist infinitely many H-unstable $T^n$-orbits when $n \geq 3$. On the other hand, we prove a monotone $T^n$-orbit in $\mathbb{C}H^n$ is H-stable and rigid for any $n$. Moreover, we see almost all Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ are not Hamiltonian volume minimizing when $n \geq 3$ as well as the case of $\mathbb{C}^n$ and $\mathbb{C}P^n$.

Inspired by the recent progress by Coates–Corti–Kasprzyk et al. on mirror symmetry for del Pezzo surfaces, we show that for any positive integer $k$ the deformation families of del Pezzo surfaces with a single $1/k(1,1)$ singularity (and no other singular points) fit into a single cascade. Additionally we construct models and toric degenerations of these surfaces embedded in toric varieties in codimension $\leq 2$. Several of these directly generalise constructions of Reid–Suzuki (in the case $k = 3$). We identify a root system in the Picard lattice, and in light of the work of Gross–Hacking–Keel, comment on mirror symmetry for each of these surfaces. Finally we classify all del Pezzo surfaces with certain combinations of $1/k(1,1)$ singularities for $k = 3,5,6$ which admit a toric degeneration.

In 1978 Uchiyama gave a proof of the characterization of $\mathrm{CMO}(\mathbb{R}^n)$ which is the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in $\mathrm{BMO}(\mathbb{R}^n)$. We extend the characterization to the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in the Campanato space with variable growth condition. As an application we characterize compact commutators $[b,T]$ and $[b,I_{\alpha}]$ on Morrey spaces with variable growth condition, where $T$ is the Calderón–Zygmund singular integral operator, $I_{\alpha}$ is the fractional integral operator and $b$ is a function in the Campanato space with variable growth condition.

We study the asymptotic behavior of metric measure spaces satisfying the rough curvature dimension condition. We prove stabilities of the rough curvature dimension condition with respect to the observable distance function and the $L^2$-transportation distance function.

We construct a framework which gives intuitive representation of local cohomology groups. By defining the concrete mappings among them, we show their equivalence. As an application, we justify intuitive representation of Laplace hyperfunctions.

Given a generic $K3$-surface $Y_k$ of the Apéry–Fermi pencil, we use the Kneser–Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T. We classify the fibrations such that the translation by T gives a Shioda–Inose structure. The other fibrations correspond to a $K3$-surface identified by its transcendental lattice. The same problem is solved for a singular member $Y_2$ of the family showing the differences with the generic case. In conclusion we put our results in the context of relations between 2-isogenies and isometries on the singular surfaces of the family.

Let $\mathbf{U}^-_q = \mathbf{U}^-_q(\mathfrak{g})$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, and $\sigma : \mathfrak{g} \to \mathfrak{g}$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak{g}^{\sigma}$ be the fixed point subalgebra of $\mathfrak{g}$, and put $\underline{\mathbf{U}}^-_q = \mathbf{U}^-_q(\mathfrak{g}^{\sigma})$. Let $\mathbf{B}$ be the canonical basis of $\mathbf{U}_q^-$ and $\underline{\mathbf{B}}$ the canonical basis of $\underline{\mathbf{U}}_q^-$. $\sigma$ induces a natural action on $\mathbf{B}$, and we denote by $\mathbf{B}^{\sigma}$ the set of $\sigma$-fixed elements in $\mathbf{B}$. Lusztig proved that there exists a canonical bijection $\mathbf{B}^{\sigma} \simeq \underline{\mathbf{B}}$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.