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Circle actions on four dimensional almost complex manifolds with discrete fixed point sets

Donghoon Jang

Abstract

We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as the Chern numbers of such an action, answering negatively a question by Sabatini whether $c_1^2[M] \leq 3 c_2[M]$ holds for any such manifold $M$. We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a $\mathbb{T}^2$-action. Furthermore, we discuss applications to circle actions on complex/symplectic 4-manifolds and semi-free circle actions with discrete fixed point sets.

Circle actions on four dimensional almost complex manifolds with discrete fixed point sets

Abstract

We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as the Chern numbers of such an action, answering negatively a question by Sabatini whether holds for any such manifold . We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a -action. Furthermore, we discuss applications to circle actions on complex/symplectic 4-manifolds and semi-free circle actions with discrete fixed point sets.

Paper Structure

This paper contains 9 sections, 34 theorems, 1 equation, 10 figures.

Table of Contents

  1. Introduction and statements of main results
  2. Background and preliminaries
  3. Classification of fixed point data in terms of multigraphs
  4. Equivariant plumbing
  5. Extendability of a graph
  6. Proofs of Theorems \ref{['t01']}, \ref{['t314']}, and \ref{['t11']}
  7. Chern numbers
  8. Comparison with complex manifolds and symplectic manifolds
  9. Semi-free actions with discrete fixed point sets

Key Result

Theorem \oldthetheorem

For $1 \leq i \leq k$, let $\{a_i,b_i\}$ be an unordered pair of relatively prime non-zero integers. There exists an effective circle action on a 4-dimensional compact, connected almost complex manifold $M$ with $k$ fixed points with weights $\{a_i,b_i\}_{i=1}^k$ if and only if the following hold. Moreover, given a pairing between the fixed points in (2a) and (2b) for each $w$ and $x$, we can cho

Figures (10)

  • Figure 1: 2-regular connected semi-free multigraph $\Gamma$ with $T(\Gamma)=2$
  • Figure 2: Operation 1
  • Figure 3: Operation 2
  • Figure 4: Operation 3
  • Figure 5: Operation 4
  • ...and 5 more figures

Theorems & Definitions (71)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 61 more