Mathematical Research Letters

Volume 18 (2011)

Number 6

Equivariant Chern numbers and the number of fixed points for unitary torus manifolds

Pages: 1319 – 1325

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n6.a19

Authors

Zhi Lü (School of Mathematical Sciences and The Key Laboratory of Mathematics for Nonlinear Sciences of Ministry of Education, Fudan University, Shanghai, 200433, China.)

Qiangbo Tan (School of Mathematical Sciences, Fudan University, Shanghai, 200433, China)

Abstract

Let $M^{2n}$ be a unitary torus $(2n)$-manifold, i.e., a $(2n)$-dimensional oriented stable complex connected closed $T^n$-manifold having a nonempty fixed point set. In this paper we show that $M$ bounds equivariantly if and only if the equivariant Chern numbers $\langle (c_1^{T^n})^i(c_2^{T^n})^j, [M]\rangle=0$ for all $i, j\in {\Bbb N}$, where $c_l^{T^n}$ denotes the $l$th equivariant Chern class of $M$. As a consequence, we also show that if $M$ does not bound equivariantly then the number of fixed points is at least $\lceil{n\over2}\rceil+1$.

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