Double point surfaces of immersions with a complex structure Seminar

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Given a self-transverse immersion f: M^{2n+2} \to R^{4n+2} of a compact closed manifold M, then its double point self-intersection manifold L^2 \to R^{4n+2} is a compact closed surface which is necessarily a boundary. If the normal bundle of the immersion f has a stable complex structure than the double point surface has a stably complex structure and so is bordant as a stably complex manifold to a multiple of CP^1 with its canonical stably complex structure. We consider what multiples can arise as double point surfaces.