An extension theorem for real Kähler submanifolds in codimension 4

Abstract

In this article, we prove a Kähler extension theorem for real Kähler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kähler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states that any real Kähler submanifolds of codimension 3 and rank at least 4 admits a Kähler extension.

A Dajczer-Rodriguez type cylinder theorem for real Kähler submanifolds

“…The symmetry relation (2.5) and (2.6) imply that ξ 2 , S = − √ −1 ξ 3 , S . So as in [23], we see that the subbundle E ⊂ T M ⊥ spanned by {ξ 2 , ξ 3 } admits an almost complex structure J, which will imply that f is partially holomorphic. This completes the proof of Theorem 3.…”

“…The terminology in (1) comes from [8] and [23], and the one in (2) was coined by Al Vitter in the context of developable submanifolds, see [20]. Each leaf of R in (3) will be called a twisted ruling for f .…”

“…We assume that either n > 4, or n ≤ 4 but ν 0 > 0. Then by [23] we have ν 0 > 0 and r ≤ 4. Theorem 1 says that f is always a twisted cylinder.…”

Holomorphicity of real Kaehler submanifolds

“…Yan and Zheng [16] observed that both results discussed above still hold under the slightly weaker assumption that the complex index of relative nullity, defined by…”

“…satisfies the same pointwise inequalities required for ν(x). The main result in [16] is that if f : M 2n → R 2n+4 satisfies ν c (x) < 2(n − 4) at any x ∈ M 2n , then there exists an open dense subset U of M 2n such that along each connected component U ′ of U, the submanifold f | U ′ has a Kaehler extension, namely, there exist a real Kaehler submanifold j : N 2n+2 → R 2n+4 and a holomorphic isometric immersion h : U ′ → N 2n+2 such that f | U ′ = j • h. Moreover, although the extension j may not be unique, it can be chosen to be minimal if f is minimal. Of course, we may be in the situation where f itself is holomorphic.…”