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Double sided torus actions and complex geometry on $SU(3)$

Hiroaki Ishida, Hisashi Kasuya

Abstract

We construct explicit complex structures and transversely Kähler holomorphic foliations on $SU(3)$ corresponding to variations of real quadratic equations on a complex quadric in $\mathbb{C}^{6}$ as generalizations of left-invariant complex structures on $SU(3) $ and an invariant Kähler structure on the flag variety $SU(3)/T$.Consequently, we obtain orbifold variants of the flag variety $SU(3)/T$ as quotients of double sided torus actions.

Double sided torus actions and complex geometry on $SU(3)$

Abstract

We construct explicit complex structures and transversely Kähler holomorphic foliations on corresponding to variations of real quadratic equations on a complex quadric in as generalizations of left-invariant complex structures on and an invariant Kähler structure on the flag variety .Consequently, we obtain orbifold variants of the flag variety as quotients of double sided torus actions.

Paper Structure

This paper contains 6 sections, 16 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Holomorphic foliations and commuting Hamiltonians
  3. $SU(3)$ as a level set
  4. Intersection of real quadrics in $M$
  5. Freeness of the double-sided action
  6. Basic and Dolbeault cohomologies

Key Result

Theorem 1.1

Let $\rho_L, \rho_R \colon (S^1)^2 \to T$ be smooth homomorphisms given by Here $w_j^L$, $w_j^R \in \mathbb{Z}^2$. Put Assume that $A_j, B_j$ for $j=1,2,3$ and $C$ satisfy Then, the following hold:

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 30 more