International Journal of Mathematics

In the present paper, we provide the foundation of a $G$-equivariant Čech–de Rham theory for a compact Lie group $G$ by using the Cartan model of equivariant differential forms. Our approach is quite elementary without referring to the Mathai–Quillen framework. In particular, by a direct computation, we give an explicit formula of the $U(l)$-equivariant Thom form of ${ℂ}^l$, which deforms the classical Bochnor–Martinelli kernel. Also, we discuss a version of equivariant Riemann–Roch formula.

In this work, we study minimal rotational surfaces in the product space ${\mathbb{Q}}_{𝜀}^2\times {𝕊}^1$, where ${\mathbb{Q}}_{𝜀}^2$ denotes either the unit $2$-sphere ${𝕊}^2$ or the $2$-dimensional hyperbolic space ${ℍ}^2$ of constant curvature $-1$, according to $𝜀=1$ or $𝜀=-1$, respectively. While there is only one kind of rotational surfaces in ${𝕊}^2\times {𝕊}^1$, there are three different possibilities for rotational surfaces in ${ℍ}^2\times {𝕊}^1$, according to the types of the induced inner product on the rotational axis of the surface. We determine the profile curves of all minimal rotational surfaces in ${\mathbb{Q}}_{𝜀}^2\times {𝕊}^1$.

We prove that wild ramification of a constructible sheaf on a surface is determined by that of the restrictions to all curves. We deduce from this result that the Euler–Poincaré characteristic of a constructible sheaf on a variety of arbitrary dimension over an algebraically closed field is determined by wild ramification of the restrictions to all curves. We similarly deduce from it that so is the alternating sum of the Swan conductors of the cohomology groups, for a constructible sheaf on a variety over a local field.

This paper deals with the curvature bound for a nonlocal curve flow with a prescribed rate of change in enclosed area via Andrews–Bryan’s distance comparison. As a by-product, a partial answer to a conjecture given by Dallaston and McCue is obtained and the $C^{\infty }$ convergence of the curvature for the nonlocal flow is achieved.

We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao’s Conjecture for arrangements having a line with at most four intersection points.

In this paper, we characterize the Hartogs domains over homogeneous Siegel domains of type II and explicitly describe their automorphism groups. Moreover, we prove that any proper holomorphic map between equidimensional Hartogs domains over homogeneous Siegel domains of type II is a biholomorphism.

We give a version of Cartan’s Theorem $A$ for nonsingular real affine varieties after blowing up.