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

TL;DR

This paper establishes a criterion based on equivariant Chern numbers to determine when a unitary torus manifold bounds equivariantly and provides a lower bound on fixed points if it does not.

Contribution

It introduces a new condition involving equivariant Chern numbers for bounding equivariantly and relates fixed point counts to this property.

Findings

A manifold bounds equivariantly if and only if all equivariant Chern numbers vanish.

If not bounding, the number of fixed points is at least loor{n/2}loor+1.

Provides a link between topological invariants and fixed point counts.

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 set. In this paper we show that $M$ bounds equivariantly if and only if the equivariant Chern numbers $< (c_1^{T^n})^i(c_2^{T^n})^j, [M]>=0$ 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 $\lceil{n\over2}\rceil+1$.