Related papers: Haupt's theorem for strata of abelian differential…

For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. Such foliations are generalizations of holomorphic principal torus bundles. If there exists a…

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the…

Finite-sheeted covering mappings onto compact connected groups are studied. It is shown that a finite-sheeted covering mapping from a connected Hausdorff topological space onto a compact connected abelian group G must be a homeomorphism…

We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies $h^G_n(X)$ over a real space manifold $X$ with $G$ a crystalline symmetry group. The…

Let $A$ be a commutative Banach algebra. Let $M$ be a complex manifold on $A$ (an $A$-manifold). Then, we define an $A$-holomorphic vector bundle $(\wedge^kT^*)(M)$ on $M$. For an open set $U$ of $M$, $\omega$ is said to be an…

We introduce an generalization of the theta divisor to the theory of holomorphic triples on a smooth projective curve $X$. We show that a given triple $T=(E_1 \to E_0)$ is $\alpha$-semistable iff there exists an orthogonal tripe $S=(F_1 \to…

In this paper, we describe an algorithm that, for a smooth connected curve $X$ over a field $k$ with normal completion having arithmetic genus $p_a(X)$, a finite locally constant sheaf $\mathcal A$ on $X_{et}$ of abelian groups of torsion…

We study finite systems of subspaces of a complex Hilbert space such that each pair of subspaces satisfies a certain condition as described in the following. For each subspace excepting the first one an angle between this subspace and the…

Assume that $A$ is a closed linear operator defined on all of a Hilbert space $H$. Then $A$ is bounded. A new short proof of this classical theorem is given on the basis of the uniform boundedness principle. The proof can be easily extended…

Let $Y$ be an abelian variety over a subfield $k \subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford-Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre…

We attempt to generalize a congruence property of elliptic modular forms proved by Sturm to that of Haupttypus of Siegel modular forms of degree 2 with level. Namely, we give an explicit bound of Fourier coefficients required to determine…

Let S be a closed surface with nonzero Euler characteristic. We prove the existence of an open neighborhood V of the identity map of S in the C^1-topology with the following property: if G is an abelian subgroup of Diff^1(S) generated by…

We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with…

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em…

Let $S$ be a Riemann surface with a non-abelian fundamental group and for each integer $k \geq 2$ or $k=\infty$, let $\widetilde{S}_{k}$ be its $k$-homology cover. The surface $\widetilde{S}_{k}$ admits a group of conformal automorphisms…

We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert…

The holomorph of a discrete group $G$ is the universal semi-direct product of $G$. In chapter 1 we describe why it is an interesting object and state main results. In chapter 2 we recall the classical definition of the holomorph as well as…

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological…

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed…

We compute (algebraically) the Euler characteristic of a complex of sheaves with constructible cohomology. A stratified Poincar\'e-Hopf formula is then a consequence of the smooth Poincar\'e-Hopf theorem and of additivity of the…