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Integral bases for the second degree cohomology of 4-dimensional toric orbifolds

Tseleung So, Jongbaek Song

Abstract

We study toric orbifolds of real dimension four with vanishing odd-degree cohomology and obtain a basis for its degree-two equivariant cohomology with integral coefficients by identifying it with the intersection of certain lattices. As applications, we provide an alternative construction of the \emph{algebraic cellular basis} for integral ordinary cohomology \cite{FSS2}. In addition, when the toric orbifold is an algebraic variety, we determine its Cartier divisor group and Picard group.

Integral bases for the second degree cohomology of 4-dimensional toric orbifolds

Abstract

We study toric orbifolds of real dimension four with vanishing odd-degree cohomology and obtain a basis for its degree-two equivariant cohomology with integral coefficients by identifying it with the intersection of certain lattices. As applications, we provide an alternative construction of the \emph{algebraic cellular basis} for integral ordinary cohomology \cite{FSS2}. In addition, when the toric orbifold is an algebraic variety, we determine its Cartier divisor group and Picard group.

Paper Structure

This paper contains 8 sections, 10 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Toric orbifolds and Weighted Stanley--Reisner rings
  3. Toric orbifolds
  4. Equivariant cohomology of $X(P, \lambda)$
  5. Intersection of lattices
  6. Applications
  7. Construction of algebraic cellular bases
  8. Description of Cartier divisors on toric surfaces

Key Result

Theorem 1.1

Let $X(P,\lambda)$ be a toric orbifold of real dimension four with trivial $H^3(X(P,\lambda);\mathbb{Z})$. As $\mathbb{Z}$-modules, it follows that where $\textbf{a},\textbf{b}$ are vectors deduced from $\lambda$, $K$ is the space of some linear relations and $\phi$ is a linear map (See eq_a_b, eq_K and eq_phi_linear_operator, respectively, for their explicit definitions). $\blacktriangleleft$$\b

Figures (1)

  • Figure 1: Polygon and Characteristic function.

Theorems & Definitions (22)

  • Theorem 1.1: restated in Theorem \ref{['thm_main']}
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 12 more