2025/04/28 by Tumpach, Alice Barbora
#53D00 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2504.19945
The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle T^*\mathbbCP(1) of the complex projective line \mathbbCP(1) ≃ \mathbbS2, which extends the Fubini-Study metric of \mathbbCP(1). By virtue of the Mostow decomposition theorem, T^*\mathbbCP(1) is isomorphic, as SU(2)-equivariant fiber bundle over \mathbbCP(1), to a complex (co-)adjoint orbit of SL(2, ℂ). In fact, this complex (co-)adjoint orbit is fibered over \mathbbCP(1)≃ \mathbbS2 with each fiber isomorphic to the hyperbolic disc ℍ2. In this paper, we are interested in the complex structure inherited on the hyperbolic disc ℍ2 by the hyperkähler extension of the 2-sphere. Contrary to what is generally believed, we show that it differs from the natural complex structure of ℍ2⊂ ℂ inherited from its embedding in ℂ. In other words, the embedding of ℍ2 with its Hermitian-symmetric structure into the hyperkähler manifold T^*\mathbbCP(1) is not holomorphic.