SU(n)-structures through quotient by torus actions

TL;DR

This paper demonstrates how $SU(n)$-structures can be obtained via symplectic quotients of Kähler manifolds with $SU(n+s)$-structures under torus actions, introducing special twist forms and exploring various geometric applications.

Contribution

It introduces the concept of twist forms to produce $SU(n)$-structures on quotients and applies this to complex projective spaces, cones over Fano manifolds, and toric bundles.

Findings

$SU(n)$-structures are inherited by symplectic quotients under certain conditions.

The existence of twist forms is crucial for the inheritance of $SU(n)$-structures.

Applications include new geometric structures on classical and toric manifolds.

Abstract

We show that if $(X,g,J,\omega)$ is a K\"ahler 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 K\"ahler-Einstein manifold and on toric $\mathbb{C}\mathbb{P}^1$ bundles. We also study the geometry behind these structures in the case of $n=3$.