Almost complex torus manifolds - a problem of Petrie type

Abstract:

The Petrie 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 in the case where the manifold admits a $T^n$-action. An almost complex torus manifold is a $2n$-dimensional compact connected almost complex manifold equipped with an effective $T^n$-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a $2n$-dimensional almost complex torus manifold $M$ only shares the Euler number with the complex projective space $\mathbb {CP}^n$, the graph of $M$ agrees with the graph of a linear $T^n$-action on $\mathbb {CP}^n$. Consequently, $M$ has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch $\chi _y$-genus, Todd genus, and signature as $\mathbb {CP}^n$, endowed with the standard linear action. Furthermore, if $M$ is equivariantly formal, the equivariant cohomology and the Chern classes of $M$ and $\mathbb {CP}^n$ also agree.