The tropical Poincaré-Hopf theorem

Poincaré duality for tropical Dolbeault cohomology of non-archimedean Mumford curves

We calculate the tropical Dolbeault cohomology for the analytifications of the projective line and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincaré duality and behaves analogously to the cohomology of curves over the complex numbers. Further, we give a complete calculation of the dimension of the cohomology on a basis of the topology.

Cycles, cocycles, and duality on tropical manifolds

We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincaré dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target.

胞腔余层的 Poincaré-Lefschetz 定理及其在 Orbifold 环面簇热带同调中的应用

A Poincaré-Lefschetz Theorem for Cellular Cosheaves and an Application to the Tropical Homology of Orbifold Toric Varieties

本文证明特定 CW 细分上胞腔余层的 Poincaré-Lefschetz 型定理，并据此将热带 Lefschetz 超平面截面定理推广至奇异热带环面簇与超曲面。

Radial index and Poincaré-Hopf index of 1-forms on semi-analytic sets

The radial index of a 1-form on a singular set is a generalization of the classical Poincaré-Hopf index. We consider different classes of closed semi-analytic sets in R^n that contain 0 in their singular locus and we relate the radial index of a 1-form at 0 on these sets to Poincaré-Hopf indices at 0 of vector fiels defined on R^n.

热带雅可比簇上的 tautological 闭链

Tautological cycles on tropical Jacobians

该研究证明了经典庞加莱公式的热带版本，同时得到带整结构实环面与热带阿贝尔簇的若干基础结果（包括热带Appell-Humbert定理），并探究了热带闭链的不同等价概念及其关联。

On Gauss-Bonnet and Poincaré-Hopf type theorems for complex $\partial$-manifolds

We prove a Gauss-Bonnet and Poincaré-Hopf type theorems for complex $\partial$-manifold $\tilde{X} = X - D$, where $X$ is a complex compact manifold and $D$ is a reduced divisor. We will consider the cases such that $D$ has isolated singularities and also if $D$ has a (not necessarily irreducible) decomposition $D=D_1\cup D_2$ such that $D_1$, $D_2$ have isolated singularities and $C=D_1\cap D_2$ is a codimension $2$ variety with isolated singularities.

指标与留数：从庞加莱-霍普夫（Poincaré-Hopf）到鲍姆-博特（Baum-Bott），以及马尔科·布鲁内拉（Marco Brunella）

Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella

该综述性文章研究向量场与全纯叶状结构的不变量，旨在关联复解析奇异簇与复流形上奇异全纯叶状结构这两个有诸多共通点的理论领域。

Lefschetz (1,1)-定理在热带几何中的版本

Lefschetz (1,1)-theorem in tropical geometry

本文证明热带流形上由热带循环基本类给出的 (n-1,n-1) 度热带同调类恰为特征波映射的核，通过建立有理多面体空间的热带 Lefschetz (1,1)-定理并结合庞加莱对偶性得出。

On indices of 1-forms on determinantal singularities

We consider 1-forms on, so called, essentially isolated determinantal singularities (a natural generalization of isolated ones), show how to define analogues of the Poincaré--Hopf index for them, and describe relations between these indices and the radial index. For isolated determinantal singularities, we discuss properties of the homological index of a holomorphic 1-form and its relation with the Poincaré--Hopf index.