Is there a multi-valued fixed point theorem for complex projective Hilbert spaces, or equivalently, Kahler manifolds with positive curvature?
In other words, an analogue of Kakutani’s theorem for complex projective Hilbert spaces. I am aware of Frankel’s theorem (https://msp.org/pjm/1961/11-1/pjm-v11-n1-p12-p.pdf) (theorem 4) which assures a fixed point for holomorphic functions from such a space to itself. Is anyone aware of work on extending Frankel’s theorem to the multi-valued case for manifolds? Thanks in advance.
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