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Kähler complexity one Hamiltonian $T$-manifolds have trivial paintings

Isabelle Charton, Liat Kessler, Susan Tolman

Abstract

Let a torus $T$ act on a symplectic manifold $(M,ω)$ with moment map $φ$. We say that the Hamiltonian $T$-manifold $(M,ω,φ)$ has complexity one if $\frac{1}{2} \dim M - \dim T = 1$, and that it is Kähler if it admits an invariant compatible complex structure. In this paper, we show how the class of Kähler complexity one Hamiltonian $T$-manifolds sits inside the class of complexity one Hamiltonian $T$-manifolds by proving that every compact, connected Kähler complexity one Hamiltonian $T$-manifold has a trivial painting. As a corollary, we show that two tall compact, connected Kähler complexity one Hamiltonian $T$-manifolds are symplectomorphic exactly if they have the same genus, Duistermaat-Heckman measure, and skeleton. Here, $(M,ω,φ)$ is tall exactly if every non-empty fiber $φ^{-1}(α)$ contains more than one orbit.

Kähler complexity one Hamiltonian $T$-manifolds have trivial paintings

Abstract

Let a torus act on a symplectic manifold with moment map . We say that the Hamiltonian -manifold has complexity one if , and that it is Kähler if it admits an invariant compatible complex structure. In this paper, we show how the class of Kähler complexity one Hamiltonian -manifolds sits inside the class of complexity one Hamiltonian -manifolds by proving that every compact, connected Kähler complexity one Hamiltonian -manifold has a trivial painting. As a corollary, we show that two tall compact, connected Kähler complexity one Hamiltonian -manifolds are symplectomorphic exactly if they have the same genus, Duistermaat-Heckman measure, and skeleton. Here, is tall exactly if every non-empty fiber contains more than one orbit.
Paper Structure (5 sections, 23 theorems, 43 equations, 2 figures)

This paper contains 5 sections, 23 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Complexity one Hamiltonian torus actions
  3. Holomorphic and Kähler torus actions
  4. The set $M_{\operatorname{core}}$
  5. Proof of Theorem \ref{['MainThm']}

Key Result

Theorem 1.1

Let $(M,\omega,\phi)$ be a compact, connected complexity one Hamiltonian $T$-manifold. If $M$ admits a $T$-invariant complex structure that is compatible with the symplectic form $\omega$, then the paintings associated to $M$ are trivial.

Figures (2)

  • Figure 4.1: The fixed points of the $S^1$-action in Example \ref{['examp:1']} and their moment images. The set $\overline{M_{\operatorname{core}}}$ is indicated by the dashed and dotted lines; the dashed black line corresponds to the sphere $N$ that is fixed by $\mathbb{Z}_2$.
  • Figure 4.2: The fixed components of the $S^1$-action in Example \ref{['examp:2']} and their moment images. The set $\overline{M_{\operatorname{core}}}$ is indicated by the dotted lines.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2: Guillemin-Sternberg, Marle
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • ...and 37 more