Almost Complex Torus Manifolds—Graphs and Hirzebruch Genera

Abstract

Abstract Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As in the case of circle actions, we show that there exists a (directed labeled) multigraph that contains information on weights at the fixed points and isotropy submanifolds of $M$. This includes the notion of a GKM (Goresky-Kottwitz-MacPherson) graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, that is, $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $\chi _y$-genus $\chi _y(M)=\sum _{i=0}^n a_i(M) \cdot (-y)^i$ of an almost complex torus manifold $M$ satisfies $a_i(M)> 0$ for $0 \leq i \leq n$. In particular, the Todd genus of $M$ is positive and there are at least $n+1$ fixed points.