For a prime number $q\neq 2$ and $r>0$, we study whether there exists an isometry of order $q^r$ acting on a free $\mathbb{Z}_{p^k}$-module equipped with a scalar product. We investigate whether there exists such an isometry with no non-zero fixed points. Both questions are completely answered in this paper if $p\neq 2,q$. As an application, we refine Naik's criterion for periodicity of links in $S^3$. The periodicity criterion we obtain is effectively computable and gives concrete restrictions for periodicity of low-crossing knots.

We study a “div-grad type” sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spectral geometric meaning when Popp's measure is considered. Our proof is probabilistic. In particular, we use Watanabe's distributional Malliavin calculus.

The goal of this article is to prove that every surface with a regular point in the three-dimensional projective space of degree at least four, is of wild representation type under the condition that either $X$ is integral or $\mathrm{Pic}(X) \cong \langle \cal{O}_X(1) \rangle$; we construct families of arbitrarily large dimension of indecomposable pairwise non-isomorphic ACM vector bundles. On the other hand, we prove that every non-integral ACM scheme of arbitrary dimension at least two, is also very wild in a sense that there exist arbitrarily large dimensional families of pairwise non-isomorphic ACM non-locally free sheaves of rank one.

In this article, we determine the automorphism groups of 14 holomorphic vertex operator algebras of central charge 24 obtained by applying the $\mathbb{Z}_2$-orbifold construction to the Niemeier lattice vertex operator algebras and lifts of the $-1$-isometries.

The Fatou–Julia decomposition is significant in the study of iterations of holomorphic mappings. Such a decomposition has been considered for foliations in a unified manner by Ghys–Gomez-Mont–Saludes, Haefliger, the author, et al. Although the decomposition will be fundamental in the study, it is not easy to determine the decomposition. In this article, we give a sufficient condition for open sets to be contained in Fatou sets. We also discuss relations between Fatou–Julia decompositions and minimal sets.

We consider supersingular abelian surfaces $A$ over a field $k$ of characteristic $p$ which are not superspecial. For any such fixed $A$, we give an explicit formula of numbers of principal polarizations $\lambda$ of $A$ up to isomorphisms over the algebraic closure of $k$. We also determine all the automorphism groups of $(A, \lambda)$ over algebraically closed field explicitly for every prime $p$. When $p \geq 5$, any automorphism group of $(A,\lambda)$ is either $\mathbb{Z}/2\mathbb{Z} = \{\pm 1\}$ or $\mathbb{Z}/10\mathbb{Z}$. When $p=2$ or 3, it is a little more complicated but explicitly given. The number of principal polarizations having such automorphism groups is counted exactly. In particular, for any odd prime $p$, we prove that the automorphism group of any generic $(A, \lambda)$ is $\{\pm 1\}$. This is a part of a conjecture by Oort that the automorphism group of any generic principally polarized supersingular abelian variety should be $\{\pm 1\}$. On the other hand, we prove that the conjecture is false for $p=2$ in case of dimension two by showing that the automorphism group of any $(A, \lambda)$ (with $\dim A = 2$) is never equal to $\{\pm 1\}$.

Lesieutre constructed an example satisfying $\kappa_{\sigma} \ne \kappa_{\nu}$. This says that the proof of the inequalities in Theorems 1.3, 1.9, and Remark 3.8 in [Fujino, On subadditivity of the logarithmic Kodaira dimension, J. Math. Soc. Japan, 69 (2017), 1565–1581] is insufficient. We claim that some weaker inequalities still hold true and they are sufficient for various applications.

There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on the complete graph. As a basic example, we compute the limiting distribution of the length of a self-avoiding walk on the complete graph, in subcritical, critical, and supercritical regimes. This provides a prototype for more complex unsolved problems such as the self-avoiding walk on the hypercube or on a high-dimensional torus.

The subject of this paper is quantum walks, which are expected to simulate several kinds of quantum dynamical systems. In this paper, we define analyticity for quantum walks on $\mathbb{Z}$. Almost all the quantum walks on $\mathbb{Z}$ which have been already studied are analytic. In the framework of analytic quantum walks, we can enlarge the theory of quantum walks. We obtain not only several generalizations of known results, but also new types of theorems. It is proved that every analytic space-homogeneous quantum walk on $\mathbb{Z}$ is essentially a composite of shift operators and continuous-time analytic space-homogeneous quantum walks. We also prove existence of the weak limit distribution for analytic space-homogeneous quantum walks on $\mathbb{Z}$.

In toric topology, to a simplicial complex $K$ with $m$ vertices, one associates two spaces, the moment-angle complex $\mathcal{Z}_K$ and the Davis–Januszkiewicz space $DJ_K$. These spaces are connected by a homotopy fibration $\mathcal{Z}_K \to DJ_K \to (\mathbb{C} P^{\infty})^m$. In this paper, we show that the map $\mathcal{Z}_K \to DJ_K$ is identified with a wedge of iterated (higher) Whitehead products for a certain class of simplicial complexes $K$ including dual shellable complexes. We will prove the result in a more general setting of polyhedral products.

This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t = F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. Under a very weak assumption we show the uniqueness of the solution of this equation. The results are applied to the problem of analytic continuation of local holomorphic solutions of equations of this type.

In this paper, as an analogue of the spectrum of a tensor-triangulated category introduced by Balmer, we define a spectrum of a triangulated category which is not necessarily tensor-triangulated. We apply it for some triangulated categories associated to a commutative noetherian ring.

We consider complex surfaces, viewed as smooth 4-dimensional manifolds, that admit hyperelliptic Lefschetz fibrations over the 2-sphere. In this paper, we show that the minimal number of singular fibers of such fibrations is equal to $2g+4$ for even $g\geq4$. For odd $g\geq7$, we show that the number is greater than or equal to $2g+6$. Moreover, we discuss the minimal number of singular fibers in all hyperelliptic Lefschetz fibrations over the 2-sphere as well.

In this paper, we classify groups which faithfully act on smooth cubic threefolds. It turns out that there are exactly 6 maximal ones and we describe them with explicit examples of target cubic threefolds.