Six-dimensional GKM manifolds with four fixed points

TL;DR

This paper classifies all 6-dimensional GKM manifolds with four fixed points, constructing examples for each type and identifying six distinct manifold types including projective spaces, quadrics, and blow-ups.

Contribution

It provides a complete classification of 6D GKM manifolds with four fixed points and constructs explicit examples for each classified type.

Findings

Six types of 6D GKM manifolds with four fixed points identified

Explicit constructions for each manifold type provided

Classification includes projective spaces, quadrics, and blow-ups

Abstract

In this paper, we study $6$-dimensional GKM manifolds with $4$ fixed points. We classify all possible GKM graphs, and for each type of graph we construct a manifold, proving the existence. We show that six types occur. (P1) complex projective space $\mathbb{C} P^3$ with standard complex structure (P2) blow up of $S^6$ at a fixed point, diffeomorphic to $\mathbb{C} P^3$ (P3) $\mathbb{C} P^3$ as the homogeneous space $\mathrm{Sp}(2)/(\mathrm{U}(1) \times \mathrm{Sp}(1))$ with non-standard almost complex structure (Q1) complex quadric $Q_3$ with standard complex structure (Q2) blow up of $S^6$ along isotropy $2$-sphere, diffeomorphic to $Q_3$ (S) $S^2 \times S^4$, obtained as equivariant gluing along orbits of two $S^6$'s