École polytechnique – Centre de Mathématiques Laurent Schwartz

10h30-12h00 – Gabriella Clemente (Université Grenoble-Alpes)

"Embedding spaces and functionals for Yau's Challenge"

Abstract: It is unknown if every compact almost complex manifold of dimension at least 3 supports an integrable almost complex structure. This talk will explore tools that could potentially detect obstructions to evolving a non-integrable structure into a hypothetical, integrable one. Any compact almost complex manifold (X,J) can be embedded into an open subset Gr of a certain Grassmannian bundle in such a way that if J is integrable, the image of the embedding is contained in a subvariety I \subset Gr. We outline a method to study the relationship between invariants of the quotient Gr/I which can be viewed as a vector bundle, and obstructions to the integrability of almost complex structures on X. We explain how in the case of S^6 this method is inconclusive if Chern classes are used as invariants. This suggests that more subtle characteristic classes are needed. And then, we turn to variational aspects of the problem. We present our findings about a family of functionals on the space of almost complex structures on X. This is part of our search for a functional that has as its critical points the integrable structures.