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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.

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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