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Hamiltonian $S^1$-manifolds which are like a coadjoint orbit of $G_2$

Hui Li

Abstract

Consider a compact symplectic manifold of dimension $2n$ with a Hamiltionan circle action. Then there are at least $n+1$ fixed points. Motivated by recent works on the case that the fixed point set consists of precisely $n+1$ isolated points, this paper studies a Hamiltonian $S^1$ action on a $10$-dimensional compact symplectic manifold with exactly 6 isolated fixed points. We study the relations of the following data: the first Chern class of the manifold,the largest weight of the action, all the weights of the action,the total Chern class of the manifold, and the integral cohomology ring of the manifold. We show how certain data can determine the others and show the similarities of these data with those of a coadjoint orbit of the exceptional Lie group $G_2$, equipped with a Hamiltonian action of a subcircle of the maximal torus.

Hamiltonian $S^1$-manifolds which are like a coadjoint orbit of $G_2$

Abstract

Consider a compact symplectic manifold of dimension with a Hamiltionan circle action. Then there are at least fixed points. Motivated by recent works on the case that the fixed point set consists of precisely isolated points, this paper studies a Hamiltonian action on a -dimensional compact symplectic manifold with exactly 6 isolated fixed points. We study the relations of the following data: the first Chern class of the manifold,the largest weight of the action, all the weights of the action,the total Chern class of the manifold, and the integral cohomology ring of the manifold. We show how certain data can determine the others and show the similarities of these data with those of a coadjoint orbit of the exceptional Lie group , equipped with a Hamiltonian action of a subcircle of the maximal torus.
Paper Structure (10 sections, 32 theorems, 179 equations)

This paper contains 10 sections, 32 theorems, 179 equations.

Table of Contents

  1. introduction
  2. the example --- a coadjoint orbit of $G_2$
  3. some basic results
  4. on the first Chern class $c_1(M)$
  5. on $c_1(M)$ and on the largest weight --- proof of Theorems \ref{['thm1']} and \ref{['thm2']}
  6. determining the ring $H^*(M; {\Bbb Z})$, all the weights and the total Chern class $c(M)$ --- proof of Theorems \ref{['thm3']} and \ref{['thm4']}
  7. Determining all the weights when the largest weight is 5 --- proof of Theorem 3 (3)
  8. Determining the sets of weights at all the fixed points --- proof of Theorem 4 (3)
  9. Determining the ring $H^*(M; {\Bbb Z})$ --- proof of Theorems 3 (1) and 4 (1)
  10. Determining the total Chern class $c(M)$ --- proof of Theorems 3 (2) and 4 (2)

Key Result

Theorem 1

Let $(M, \omega)$ be a compact $10$-dimensional Hamiltonian $S^1$-manifold. Assume $M^{S^1}=\{P_0, P_1, \cdots, P_5\}$ and $[\omega]$ is a primitive integral class. Then $c_1(M)=k[\omega]$ with $k\in{\Bbb N}$ and $1\leq k\leq 6$, and the largest weight is at least 5.

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 52 more