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Hello,

I would like to know some examples of totally complex submanifolds in quaternionic Kaehlerian manifolds, is there any references in which I could find them? So far I could not find. Also, the interesting example would be a holomorphic submanifold that is not totally complex in quaternionic Kaehlerian manifolds.

Thank you in advance!

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  • $\begingroup$ What do you mean by totally complex? Remember, a quaternionic-Kahler manifold is not necessarily a complex manifold. [If you include hyperKahler manifolds in the class of quaternionic-Kahler manifolds, which is a matter of convention, then those would of course be complex.] Do you mean a submanifold which admits an integrable almost complex structure? If that is what you mean, what would the complex structure have to do with the ambient quaternionic-Kahler structure? $\endgroup$
    – Spiro Karigiannis
    Commented Apr 19, 2012 at 13:28
  • $\begingroup$ I was thinking on this: On an ambient manifold M we have 3 almost complex structures, J, K and L. And if for a submanifold N these conditions are satisfied: 1) $\nabla_{X}J=0$, for $X\in T(N)$, 2) $J(T(M))=T(M)$, $K(T(M))\bot T(M)$, $L(T(M))\bot T(M)$, then we say that N is totally complex submanifold. Or maybe I misunderstood it. $\endgroup$
    – Mirjana
    Commented Apr 20, 2012 at 10:01
  • $\begingroup$ Well, you got two nice answers from Robert Bryant and Nina. But I wanted to add the following comment: The structures J, K, L that you mention are not globally well-defined on a quaternionic-Kahler manifold, unless it is hyperKahler. They are only locally defined. They give rise to (again, locally defined) Kahler forms $\omega_J$, $\omega_K$, and $\omega_L$, but the 4-form $\Phi = \omega_J^2 + \omega_K^2 + \omega_L^2$ is globally well-defined. $\endgroup$
    – Spiro Karigiannis
    Commented Apr 20, 2012 at 14:49
  • $\begingroup$ @Mirjana: Is your definition of 'holomorphic' just a submanifold $N^{2k}\subset M^{4n}$ such that there exists a section $J:N\to Q$ so that $T_xN$ is a $J_x$-complex subspace for all $x\in N$, or do you also require that the section $J$ define an integrable almost complex structure on $N$? (Note that a given $N$ might have more than two sections $J$ with these properties, but this is not common.) $\endgroup$
    – Robert Bryant
    Commented Apr 20, 2012 at 17:18
  • $\begingroup$ @Robert: I don't think than J should be integrable. I wonder does the condition 1) from my previous comment and the condition J(T(M))=T(M) necessarily imply that $K(T(M))\bot T(M)$ and $L(T(M))\bot T(M)$, for other two sections. So, holomorphic submanifold should satisfy just the condition 1) and J(T(M)) = T(M). But, the other two conditions for K and L don't have to be satisfied. Let's take on simple example: let $\phi:C\to C^{2}$ be an immersion. $\Omega = dz\wedge dw$ is standard complex symplectic form. $z=x+iu$; $w=y+iv$. Let $\tau$ be the standard real structure and its fixed point set $\endgroup$
    – Mirjana
    Commented Apr 21, 2012 at 2:34

3 Answers 3

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These examples are easy to come by: Let $Q\to M$ be the canonical twistor bundle over the Q-K manifold $M^{4n}$. Then $Q$ is a holomorphic contact manifold of complex dimension $2n{+}1$. Now let $L\subset Q$ be any Legendrian (or sub-Legendrian) holomorphic submanifold and project it into $M$. This will be a manifold of the desired type.

For example, if $M$ is quaternionic projective space, then $Q$ is a complex projective space, and there are many algebraic Legendrian submanifolds.

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I am not sure that projecting a legendrian submanifold in $Q$ to $M$ you get again a submanifold, many singularities can occur. For example if you take one of the simplest inhomogeneous Legendrian submanifolds considered by Buczynski in "HYPERPLANE SECTIONS OF LEGENDRIAN SUBVARIETIES", (if I remember correctly) or in Landsberg Manivel "Legendrian varieties", it seems to me that the projection is not smooth.

Several examples of Totally complex submanifolds can be found in:

M. Takeuchi: Totally complex submanifolds of quaternionic symmetric spaces, Japan. J. Math., (N.S.) 12 (1986) 161–189.

K. Tsukada: Parallel submanifolds in a Quaternion projective space, Osaka J. Math 22 (1985), 187–241.

K. Tsukada: Einstein-K ̈ahler submanifolds in a quaternion projective space, Bull. London Math. Soc. 36 (2004), 527–536.

Bedulli-Gori-podestà "Maximal totally complex submanifolds of $\mathbb{H}\mathbb{P}^n$: homogeneity and normal holonomy" Bullettin of LMS 41 (2009) 10029--1040\

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  • $\begingroup$ @Nina: 1. Mirjana didn't ask for compact examples or examples of maximal dimension, just for examples, of which there are plenty (local ones and of complex dimension at most $n$ when $M$ has (real) dimension $4n$), as my answer indicates. 2. You are mistaken about projections from $Q$ to $M$ introducing image singularities (other than self-intersections) for complex Legendrian submanifolds of $Q$; any Legendrian submanifold of $Q$ is transverse to the fibers. 3. It is true that many (sub-)Legendrian varieties in $Q$ are singular, but many are not, particularly ones with low dimension. $\endgroup$
    – Robert Bryant
    Commented Apr 20, 2012 at 16:25
  • $\begingroup$ @Nina: Thank you for the references! $\endgroup$
    – Mirjana
    Commented Apr 21, 2012 at 1:09
  • $\begingroup$ @Robert: Yes you are right, I have to say "I am not sure that ALWAYS projecting a Legendrian submanifold in Q to M you get a submanifold." In the examples that I have in mind the singularities (self-intersections) occur. $\endgroup$
    – Nina
    Commented Apr 21, 2012 at 13:39
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I think this paper is also useful: arXiv:math/0308283 Complex Forms of Quaternionic Symmetric Spaces by Joseph A. Wolf. There are also several papers by Alekseevsky and Marchiafava. Liviu

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