Almost complex torus manifolds -- graphs, Hirzebruch genera, and problem of Petrie type · arXivDesk
2201.00352Jan 2, 2022Changed from almost complex toric manifolds to almost complex torus manifolds, to avoid confusion with toric manifolds that are complex. Minor revisions
Almost complex torus manifolds -- graphs, Hirzebruch genera, and problem of Petrie type
Let a k-dimensional torus Tk 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
Nearby in the stack
. 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
χy
-genus
χy(M)=∑i=0nai(M)⋅(−y)i
of an almost complex torus manifold
M
satisfies
ai(M)>0
for
0≤i≤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
CPn
admits a non-trivial circle action, its Pontryagin class agrees with that of
CPn
. Petrie proved this conjecture if instead it admits a
Tn
-action. We prove that if a
2n
-dimensional almost complex torus manifold
M
only shares the Euler number with the complex projective space
CPn
, an associated graph agrees with that of a linear
Tn
-action on
CPn
; consequently
M
has the same weights at the fixed points, Chern numbers, equivariant cobordism class, Hirzebruch
χy
-genus, Todd genus, and signature as
CPn
. If furthermore
M
is equivariantly formal, the equivariant cohomology and the Chern classes of