Geodesic
Geometry of universal embedding spaces for almost complex manifolds
Archivum mathematicum, Tome 60 (2024) no. 1, pp. 35-60

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We investigate the geometry of universal embedding spaces for compact almost-complex manifolds of a given dimension, and related constructions that allow for an extrinsic study of the integrability of almost-complex structures. These embedding spaces were introduced by J-P. Demailly and H. Gaussier, and are complex algebraic analogues of twistor spaces. Their goal was to study a conjecture made by F. Bogomolov asserting the “transverse embeddability” of arbitrary compact complex manifolds into foliated algebraic varieties. In this work, we introduce a more general category of universal embedding spaces, and elucidate the geometric structure of related bundles, such as the integrability locus characterizing integrable almost-complex structures. Our approach could potentially lead to finding new obstructions to the existence of a complex structure, which may be useful for tackling Yau’s Challenge.
DOI : 10.5817/AM2024-1-35
Classification : 32L05, 32Q40, 32Q60
Keywords: almost-complex manifolds; complex structures; integrability; Nijenhuis tensor; obstruction theory; transverse embeddings; fiber bundles; vector bundles 
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Clemente, Gabriella. Geometry of universal embedding spaces for almost complex manifolds. Archivum mathematicum, Tome 60 (2024) no. 1, pp. 35-60. doi: 10.5817/AM2024-1-35

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