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# Holomorphicity of real Kaehler submanifolds

## Sergio Chion Aguirre IMPA, 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

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