Toroidal b-divisors and Monge–Ampère measures

We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of nef b-divisors on complete varieties which are toroidal with respect to a snc divisor. As a key ingredient we show the existence of a limit measure, supported on a balanced rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can be computed as an integral with respect to this limit measure. As an application, we show that a Hilbert–Samuel type formula holds for big and nef toroidal Weil b-divisors.

Ramification conjecture and Hirzebruch’s property of line arrangements

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on $\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a $\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is $\text{CAT}[0]$ if the metric on $\mathbb{C}\text{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in $\mathbb{C}\text{P}^{2}$ studied by Hirzebruch have aspherical complement.

Polyhedral Space Curve Meshing Reducer With Multiple Output Shafts

In recent years, a gear named Space Curve Meshing Wheel (SCMW) has been invented based on the meshing theory of space curves instead of classic space surfaces. Well improved in many aspects after its invention, it has been applied within the Space Curve Meshing Reducer (SCMR). The design method of an invention named polyhedral SCMR is presented in this paper. With single input shaft and multiple output shafts, this SCMR has advantages like compact structure, flexible design and low cost. It is characterized by the application of the SCMW group containing one driving wheel and several driven wheels, whose rotation axes are concurrent at a point and radiate in polyhedral directions. A SCMW group can form a single-stage SCMR, while SCMW groups connected can form a multiple-stage SCMR. In this paper, geometric parameters of the polyhedral SCMR are defined, design formulas are derived, and an example is provided to illustrate the design process.