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Keywords:
almost-complex manifolds; complex structures; integrability; Nijenhuis tensor; obstruction theory; transverse embeddings; fiber bundles; vector bundles
almost-complex manifolds; complex structures; integrability; Nijenhuis tensor; obstruction theory; transverse embeddings; fiber bundles; vector bundles
Summary:
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.
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