Almost complex torus manifolds - a problem of Petrie type

The Petrie conjecture asserts that if a homotopy C P n \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of C P n \mathbb {CP}^n . Petrie proved this conjecture in the case where the manifold admits a T n T^n -action. An almost complex torus manifold is a 2 n 2n -dimensional compact connected almost complex manifold equipped with an effective T n 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 2 n 2n -dimensional almost complex torus manifold M M only shares the Euler number with the complex projective space C P n \mathbb {CP}^n , the graph of M M agrees with the graph of a linear T n T^n -action on C P n \mathbb {CP}^n . Consequently, M M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch χ y \chi _y -genus, Todd genus, and signature as C P n \mathbb {CP}^n , endowed with the standard linear action. Furthermore, if M M is equivariantly formal, the equivariant cohomology and the Chern classes of M M and C P n \mathbb {CP}^n also agree.