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SU(n)-structures through quotient by torus actions

Quentin Peres

TL;DR

The paper develops a twist-based quotient construction that starts from a Kähler manifold with an $SU(n+s)$-structure and a Hamiltonian holomorphic $T^s$-action, and yields an $SU(n)$-structure on the symplectic quotient provided a twist form exists with charge $\tfrac{q}{2}$, formalized through descended $\Omega$ and $\omega$ and their wedge relations. It develops both a single-twist reduction and a multi-twist extension to handle families of orthogonal twist forms, enabling descent of the $SU(n)$-structure to quotients under broader conditions. The authors apply the framework to explicit settings, including holomorphic twist forms on $\mathbb{C}^N$, construction of $SU(3)$-structures on $\mathbb{C}\mathbb{P}^3$ and $SU(4n-1)$-structures on $\mathbb{C}\mathbb{P}^{4n-1}$ via Gram–Schmidt, toric $\mathbb{C}\mathbb{P}^1$-bundles over Hirzebruch surfaces, and Calabi–Yau cones over Fano Kähler–Einstein manifolds. They also analyze torsion for the CP$^3$ twist-structured families, identifying when the resulting structures are LT and detailing singularities arising in multi-twist constructions. This framework broadens the toolkit for constructing $SU(n)$-structures with torsion on quotients of Kähler manifolds, with potential implications for string compactifications and geometric analysis.

Abstract

We show that if $(X,g,J,ω)$ is a Kähler manifold with an $SU(n+s)$-structure and a Hamiltonian holomorphic action of a compact torus $T^s$, then the usual symplectic quotient $Y$ inherits an $SU(n)$-structure provided the existence of special $1$-forms on $X$, called twist forms. We then give several applications of our results: on complex projective spaces, on cones over Fano Kähler-Einstein manifold and on toric $\mathbb{C}\mathbb{P}^1$ bundles. We also study the geometry behind these structures in the case of $n=3$.

SU(n)-structures through quotient by torus actions

TL;DR

The paper develops a twist-based quotient construction that starts from a Kähler manifold with an -structure and a Hamiltonian holomorphic -action, and yields an -structure on the symplectic quotient provided a twist form exists with charge , formalized through descended and and their wedge relations. It develops both a single-twist reduction and a multi-twist extension to handle families of orthogonal twist forms, enabling descent of the -structure to quotients under broader conditions. The authors apply the framework to explicit settings, including holomorphic twist forms on , construction of -structures on and -structures on via Gram–Schmidt, toric -bundles over Hirzebruch surfaces, and Calabi–Yau cones over Fano Kähler–Einstein manifolds. They also analyze torsion for the CP twist-structured families, identifying when the resulting structures are LT and detailing singularities arising in multi-twist constructions. This framework broadens the toolkit for constructing -structures with torsion on quotients of Kähler manifolds, with potential implications for string compactifications and geometric analysis.

Abstract

We show that if is a Kähler manifold with an -structure and a Hamiltonian holomorphic action of a compact torus , then the usual symplectic quotient inherits an -structure provided the existence of special -forms on , called twist forms. We then give several applications of our results: on complex projective spaces, on cones over Fano Kähler-Einstein manifold and on toric bundles. We also study the geometry behind these structures in the case of .

Paper Structure

This paper contains 13 sections, 17 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. A twist construction
  3. Existence and space of twist forms
  4. Explicit examples and torsion
  5. A twist reduction
  6. An abstract version of the reduction process of Larfors,Tsimpis and Lüst
  7. A multiple twist theorem
  8. Geometric applications
  9. Twist forms on $(\mathbb{C}^N,(\mathbb{C}^*)_{diag})$
  10. Case of $\mathbb{C}\mathbb{P}^{4n-1}$
  11. Revisiting $SU(3)$-structures on toric $\mathbb{C}\mathbb{P}^1$-bundles
  12. Calabi-Yau cones over Fano Kähler-Einstein manifolds
  13. Torsion of $SU(3)$-structures on $\mathbb{C}\mathbb{P}^3$

Key Result

Theorem 1

Let $(X,h_X,J_X,\omega_X,\Omega_X)$ be a Kähler manifold of complex dimension $n+s$ with $(\Omega_X,\omega_X)$ an $SU(n+s)$-structure on $X$ with charge operator $q$ taking values in the pure imaginary numbers. Consider a free, Hamiltonian and holomorphic action of a compact torus $T^s$ on $X$ with descend to $Y$ and satisfy the equations: where $\overline{\alpha_M}\cdot$ is the contraction of f

Theorems & Definitions (33)

  • Definition 1
  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • Proposition 7
  • Theorem 8
  • Theorem 9
  • ...and 23 more