International Journal of Mathematics

We show that for any positive integer $n\ge 2$, there exist pairs of compact, contractible, Stein 4-manifolds and order $n$ self-diffeomorphisms of the boundaries that do not extend to the full manifolds. Each boundary of the Stein 4-manifolds is a cyclic branched cover along a slice knot embedded in the boundary of a contractible 4-manifold. Each pair is called a finite order cork, we give a method producing examples of many finite order corks, which are possibly not a Stein manifold. The example of the Stein cork gives a diffeomorphism generating $n$ homotopic but non-isotopic Stein fillable contact structures for an arbitrary positive integer $n$.

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on $\lambda$-invariants in a $p$-extension of ${\mathbb{Z}}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched ${\mathbb{Z}}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

Let $C$ be a smooth complex projective curve of genus $g\ge 2$ and let $p\in C$ be a point. From Hecke correspondence, any stable bundle on $C$ of rank $r$ and determinant $O_C(p)$ defines a rational family of semistable vector bundles on $C$ of rank $r$ and trivial determinant. In this paper, we study linear systems of theta divisors associated to these families.

We construct a complex algebraic surface with geometric genus $p_g=3$, irregularity $q=0$, self-intersection of the canonical divisor $K^2=24$ and canonical map of degree $24$ onto ${\mathbb{P}}^2$.

Suppose $K$ is a finite field extension of ${\mathbb{Q}}_p$ containing a primitive $p$th root of unity. Let $K_{<p}$ be a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$. In this paper, we develop formalism which allows us to study the structure of ${\mathrm{\Gamma }}_{<p}=\mathrm{\text{Gal}}(K_{<p}/K)$ via methods of Lie theory. In particular, we introduce an explicit construction of a Lie ${𝔽}_p$-algebra $L$ and an identification ${\mathrm{\Gamma }}_{<p}=G(L)$, where $G(L)$ is a $p$-group obtained from the elements of $L$ via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration ${\{{\mathrm{\Gamma }}_{<p}^{(v)}\}}_{v\ge 0}$ and an explicit form of the Demushkin relation for ${\mathrm{\Gamma }}_{<p}$.

In this paper, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singular cases are also investigated.

We give a variant of the Beauville–Narasimhan–Ramanan correspondence for irregular parabolic Higgs bundles on smooth projective curves with fixed semi-simple irregular part and show that it defines a Poisson isomorphism between certain irregular Dolbeault moduli spaces and relative Picard bundles of families of ruled surfaces over the curve.

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and $\mathrm{\Gamma }$ a discontinuous group for the homogeneous space $\mathcal{𝒳}=G/H$. Given a deformation parameter $\phi \in \mathrm{\text{Hom}}(\mathrm{\Gamma },G)$, the deformed subgroup $\phi (\mathrm{\Gamma })$ may fail to act properly discontinuously on $\mathcal{𝒳}$. To understand this phenomenon in the case when $G$ stands for an Euclidean motion group $O_n(ℝ)⋉{ℝ}^n$, we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of ${\mathbb{Z}}^k$ on ${ℝ}^{k+1}$, Int. J. Math. 17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when $\mathrm{\Gamma }$ turns out to be a crystallographic subgroup of $G$.

Let $X$ be a smooth projective toric variety with $k$ ample line bundles. Let $Z$ be the zero locus of $k$ generic sections. It is well known that the ambient quantum ${𝒟}$-module of $Z$ is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of $X$ and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum ${𝒟}$-module.

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $ℝ$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$-bundles on $X$ determine uniquely the isomorphism class of $X$.