Almost complex torus manifolds -- graphs, Hirzebruch genera, and problem of Petrie type

Abstract

Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As for circle actions, we show that there exists a (directed labeled) multigraph that encodes weights at the fixed points of $M$. This includes the notion of a GKM graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, i.e., $M$ 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. Petrie's conjecture asserts that if a homotopy $\mathbb{CP}^n$ admits a non-trivial circle action, its Pontryagin class agrees with that of $\mathbb{CP}^n$. Petrie proved this conjecture if instead it admits a $T^n$-action. We prove that if a $2n$-dimensional almost complex torus manifold $M$ only shares the Euler number with the complex projective space $\mathbb{CP}^n$, an associated graph agrees with that of a linear $T^n$-action on $\mathbb{CP}^n$; consequently $M$ has the same weights at the fixed points, Chern numbers, equivariant cobordism class, Hirzebruch $\chi_y$-genus, Todd genus, and signature as $\mathbb{CP}^n$. If furthermore $M$ is equivariantly formal, the equivariant cohomology and the Chern classes of $M$ and $\mathbb{CP}^n$ also agree.