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Properties of a hypothetical exotic complex structure on $$\mathbb C\mathbb P^3$$. (English) Zbl 1174.53345
Summary: We consider almost-complex structures on $$\mathbb C\mathbb P^3$$ whose total Chern classes differ from that of the standard (integrable) almost-complex structure. E. Thomas established the existence of many such structures. We show that if there exists an “exotic” integrable almost-complex structures, then the resulting complex manifold would have specific Hodge numbers which do not vanish. We also give a necessary condition for the nondegeneration of the Frölicher spectral sequence at the second level.

##### MSC:
 53C56 Other complex differential geometry 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58J20 Index theory and related fixed-point theorems on manifolds 55T99 Spectral sequences in algebraic topology
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