Circle actions on unitary manifolds with discrete fixed point sets

TL;DR

This paper investigates circle actions on compact unitary manifolds with discrete fixed points, establishing relationships between fixed point weights, encoding data via multigraphs, and deriving bounds on fixed points and genus.

Contribution

It generalizes results from almost complex manifolds to unitary manifolds, introduces multigraph encoding of fixed point data, and provides bounds on fixed points and genus for specific actions.

Findings

Relationships between weights at fixed points are established.

A multigraph encoding fixed point data is constructed.

Lower bounds on fixed points and the Hirzebruch genus are derived.

Abstract

In this paper, we prove various results for circle actions on compact unitary manifolds with discrete fixed point sets, generalizing results for almost complex manifolds. For a circle action on a compact unitary manifold with a discrete fixed point set, we prove relationships between the weights at the fixed points. As a consequence, we show that there is a multigraph that encodes the fixed point data (a collection of multisets of weights at the fixed points) of the manifold; this can be used to study unitary $S^1$-manifolds in terms of multigraphs. We derive results regarding the first equivariant Chern class, obtaining a lower bound on the number of fixed points under an assumption on a manifold. We determine the Hirzebruch $\chi_y$-genus of a compact unitary manifold admitting a semi-free $S^1$-action, and obtain a lower bound on the number of fixed points.