A note on torus actions and the Witten genus

Abstract

We show that the Witten genus of a string manifold $M$ vanishes, if there is an effective action of a torus $T$ on $M$ such that $\dim T>b_2(M)$. We apply this result to study group actions on $M\times G/T$, where $G$ is a compact connected Lie group and $T$ a maximal torus of $G$. Moreover, we use the methods which are needed to prove these results to the study of torus manifolds. We show that up to diffeomorphism there are only finitely many quasitoric manifolds $M$ with the same cohomology ring as $#_{i=1}^k \pm\mathbb{C} P^n$ with $k<n$.

10 pages; presentation improved; to appear in Pacific J. Math

References in corpus (2)

• Torus manifolds and non-negative curvature
• Equivariantly homeomorphic quasitoric manifolds are diffeomorphic

Cited by in corpus (3)

• On a conjecture of Stolz in the toric case
• Witten Genus on String Toric Complete Intersections
• Witten genera of complete intersections