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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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