Maximally non-integrable almost complex structures: an h-principle and cohomological properties

R. Coelho ^[2] ; G. Placini ^[1] ; J. Stelzig ^[2]
1. [1] Ludwig Maximilian University of Munich
  
  Ludwig Maximilian University of Munich
  
  Kreisfreie Stadt München, Alemania
2. [2] Università di Cagliari, Italy
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 5, 2022
Idioma: inglés
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Resumen
- We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an h-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension 2n ≥ 10 (respectively ≥ 6) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed 4-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds).