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# Holomorphicity of real Kaehler submanifolds

## Sergio Chion AguirreIMPA, Río de Janeiro

I will discuss the subject of real Kaehler submanifolds, that is, isometric immersions $$f\colon M^{2n}\to\mathbb{R}^{2n+p}$$ of a Kaehler manifold $$(M^{2n},J)$$ of complex dimension $$n\geq 2$$ into Euclidean space with codimension $$p$$. In particular, I will present a recent result that shows that for codimension $$2p\leq 2n-1$$ generic rank conditions on the second fundamental form of $$f$$ imply that the submanifold has to be minimal. In fact, for codimension $$p\leq 11$$ we have a stronger conclusion, namely, that $$f$$ must be holomorphic with respect to some complex structure in the ambient space.

This is joint work with A. de Carvalho and M. Dajczer.

# Sergio Chion Aguirre

## IMPA, Río de Janeiro (Brasil)

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