We investigate U(1)-equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all U(1)-subgroups of the torus for which one can obtain U(1)-equivariant deformation that do not preserve semi-free U(1)-action. This gives many new self-dual metrics with U(1)-action which are not conformally isometric to LeBrun metric. We also count the dimension of the moduli space of self-dual metrics with U(1)-action obtained in this way.
Generalized geometry, equivariant $\bar{\partial}\partial$-lemma, and torus actions
Toric geometry of $G_2$-manifolds
On a construction of the twistor spaces of Joyce metrics, II
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