Conification of K\"ahler and hyper-K\"ahler manifolds

TL;DR

This paper constructs conical K"ahler and hyper-K"ahler manifolds from given manifolds with specific Killing vector fields, linking these geometric structures to supergravity c-maps and extending Haydys' theorem.

Contribution

It introduces a method to generate conical manifolds from K"ahler and hyper-K"ahler manifolds with particular symmetries, relating to supergravity c-maps and generalizing Haydys' results.

Findings

Constructed conical K"ahler manifolds via Hamiltonian Killing vectors.

Built hyper-K"ahler cones from manifolds with specific Killing vector properties.

Connected the geometric constructions to supergravity c-map and scalar curvature.

Abstract

Given a K\"ahler manifold $M$ endowed with a Hamiltonian Killing vector field $Z$, we construct a conical K\"ahler manifold $\hat{M}$ such that $M$ is recovered as a K\"ahler quotient of $\hat{M}$. Similarly, given a hyper-K\"ahler manifold $(M,g,J_1,J_2,J_3)$ endowed with a Killing vector field $Z$, Hamiltonian with respect to the K\"ahler form of $J_1$ and satisfying $\mathcal{L}_ZJ_2= -2J_3$, we construct a hyper-K\"ahler cone $\hat{M}$ such that $M$ is a certain hyper-K\"ahler quotient of $\hat{M}$. In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-K\"ahler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-K\"ahler cone, which in turn defines a quaternionic K\"ahler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.