Address: IHES, 35 route de Chartres, Bures-sur-Yvette, F-91440, France
E-mail: clemente@ihes.fr
Abstract: 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.
AMSclassification: primary 32Q60; secondary 32L05, 32Q40.
Keywords: almost-complex manifolds, complex structures, integrability, Nijenhuis tensor, obstruction theory, transverse embeddings, fiber bundles, vector bundles.
DOI: 10.5817/AM2024-1-35