Six-dimensional GKM manifolds with four fixed points

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