A TWISTORIAL CONSTRUCTION OF (1, 1)-geodesic MAPS INTO COMPLEX GRASSMANNIANS

PIOTR KOBAK

IMUJ PREPRINT 2003/09

Abstract. (1, 1)-geodesic immersions into complex Grassmann spaces Gr_{p}(ℂ^{n}) are constructed with the aid of twistor fibrations. This provides examples of (1, 1)-geodesic submanifolds of maximal dimension in complex Grassmannians, thus generalising constructions known for Gr_{2}(ℂ^{n}). It is also shown that, except for the Gr_{2}(ℂ^{n}) case, all full isotropic (1, 1)-geodesic immersions arise from Grassmann bundles over holomorphic curves via this construction.

1. Introduction

Consider a map φ from a Kähler manifold M into a Riemannian manifold N. If the restriction of φ to every holomorphic curve in M is harmonic then φ is called pluriharmonic. A pluriharmonic map which is an isometry is called (1,1)-geodesic, since the condition for pluriharmonicity is given by the equation (∇dφ)^{(1,1)} = 0 (see [14]). It is also worth pointing out that (1, 1)-geodesic submanifolds provide examples of austere submanifolds, considered by Bryant in [3].

If N is a symmetric space and φ : M → N is a nontrivial pluriharmonic immersion then the dimension of M cannot exceed the value of a certain invariant p(N); for the complex Grassmann manifold Gr_{p}(ℂ^{p+q}) this is equal to (p − 1)(q − 1) + 1. By nontrivial we mean that φ is not ±holomorphic. We refer the reader to [9] for more background information and further references. Constructions of minimal surfaces in complex Grassmannians have been known for some time, see for example [6, 5] and the references therein. In [9] they were generalised to higher dimension, giving examples of nontrivial compact (1,1)-geodesic submanifolds in complex Grassmannians. These submanifolds reach the maximal possible dimension if p = 2 (a quaternionKähler version of this case is discussed in [11, 12, 13]).

The aim of this paper is to present a new construction of (1, 1)-geodesic submanifolds in complex Grassmann spaces. It provides compact submanifolds of the maximal dimension in Gr_{p}(ℂ^{n}), with no restriction on p. The idea is to exploit twistor fibrations F → Gr_{p}(ℂ^{n}) of flag manifolds over Grassmann spaces. A twistor bundle has a natural horizontal distribution, and horizontal holomorphic curves project to harmonic surfaces in the base. Consequently, horizontal holomorphic submanifolds in F project to (1,1)-geodesic submanifolds in Gr_{p}(ℂ^{n}). We consider flag manifolds F = F_{1,q+1,n} which parametrise lines in (q + 1)-spaces in ℂ^{n}. They fibre holomorphically over ℂℙ^{n−1}, and horizontal holomorphic submanifolds in F are constructed from normal bundles of linearly full holomorphic curves in ℂℙ^{n−1}. Although there are many types of flag manifolds fibering twistorialy over Gr_{p}(ℂ^{n}) (cf. [5, 4]), we show that (except for Gr_{2}(ℂ^{n})) one has to use F_{1,q+1,n} in order to get maximal (1,1)-geodesic submanifolds in Gr_{p}(ℂ^{n}).

2. Complex Grassmannians and Flag manifolds

2.1. Basic properties. Flag manifolds are generalisations of projective spaces and Grassmann manifolds. They can be defined in abstract manner for any complex semisimple Lie group as conjugacy classes of parabolic algebras. For the general theory of twistor fibrations for symmetric spaces, and applications to harmonic maps, see [4]. In the case of unitary groups there is a more intuitive geometric description. We recall here only the basic facts on unitary flag manifolds, their tangent spaces, and twistor fibrations. More details can be found for example in [6]; the tournament approach [5] is also worth mentioning.

For each sequence of integers 0 = ℓ_{0} < ℓ_{1} < … < ℓ_{m} = n the flag manifold F = F_{ℓ_{1},ℓ_{2},...,ℓ_{m}} consists of all sequences of nested vector spaces {0} = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ … ⊂ V_{m} = ℂ^{n} where dim V_{i} = ℓ_{i}. It is a homogeneous space, and can be written as

F = Gl(n, ℂ)/P

where P is a parabolic group. Fix a Hermitian form on ℂ^{n}. One finds that F is also homogeneous under the compact group action:

F = \frac{SU(n)}{S (U(l_{1}) × U(l_{2}) × … × U(l_{m}))}

where

l_{i} = ℓ_{i} − ℓ_{i−1}, i = 1 … m.

This corresponds to the geometric description of flag manifolds as spaces of orthogonal decompositions of ℂ^{n} :

F ≃ F_{l_{1},l_{2},...,l_{m}}
= {(V_{1}, V_{2}, …, V_{m}) : V_{1} ⊕ … ⊕ V_{m} = ℂ^{n},
V_{i} ⊥ V_{j} for i ≠ j, dim V_{i} = l_{i}}

where V_{i} is the orthogonal complement of V_{i−1} in V_{i}. Note that V_{i} = V_{1}⊕V_{2}⊕…⊕V_{i}. In this paper both descriptions are used. For clarity we reserve calligraphic font for holomorphic flags while orthogonal flags are typeset in standard font.

If v = (V_{1}, V_{2}, …, V_{m}) is an element of F_{l_{1},...l_{m}} then the complexified tangent space T_{v}^{ℂ}F_{l_{1},...l_{m}} can be written as

T_{v}^{ℂ}F_{l_{1},...l_{m}} = ⨁_{i≠j} Hom(V_{i}, V_{j})

and the canonical holomorphic structure on F_{l_{1},...l_{m}} is determined by the splitting T^{ℂ} = T^{1,0} ⊕ T^{0,1}, where

T_{v}^{1,0}F_{l_{1},...l_{m}} = ⨁_{i<j} Hom(V_{i}, V_{j}), T_{v}^{0,1}F_{l_{1},...l_{m}} = ⨁_{i>j} Hom(V_{i}, V_{j}).

Under the F_{ℓ_{1},ℓ_{2},...,ℓ_{m}} ≃ F_{l_{1},...l_{m}} identification T^{1,0}F_{l_{1},...l_{m}} is the holomorphic tangent space of F_{ℓ_{1},ℓ_{2},...,ℓ_{m}}: let γ be a holomorphic curve ℂ ∋ z ↦ (V_{1}(z), …, V_{m}(z)) ∈ F_{ℓ_{1},ℓ_{2},...,ℓ_{m}}. Then the Hom(V_{i}, V_{j}) component of a vector tangent to γ is the map

V_{i} ∋ s → π_{j} (\frac{d}{dz} σ_{s}(z))_{z=0} ∈ V_{j}  (1)

where σ_{s} : ℂ ∋ z ↦ ℂ^{n} is an arbitrary holomorphic map such that σ_{s}(z) ∈ V_{i}(z), σ_{s}(0) = s; here π_{j} : ℂ^{n} → V_{j} is the orthogonal projection. In other words the map defined in (1) is a component of the fundamental form of the bundle γ_{∗}V_{i} (viewed as a subbundle of the trivial bundle ℂ × ℂ^{n}).

Let p = ∑_{i odd} l_{i}. The fibre bundle

ζ : F_{l_{1},l_{2},...,l_{m}} ∋ (V_{1}, V_{2}, …, V_{m}) ↦ ⨁_{i odd} V_{i} ∈ Gr_{p}(ℂ^{n}).  (2)

is called a twistor fibration.

Remark 1. This is the ,,Grassman tournament” definition as in [5]. We will assume that m ≥ 3 so that ζ is nontrivial (in which case it is neither holomorphic nor anitiholomorphic). One should also consider twistor projections defined by sums of even flags, V_{2}⊕V_{4}⊕…, but these can be obtained by composing the odd fibrations with the antiholomorphic isometry

Gr_{p}(ℂ^{n}) ∋ v ↦ v^{⊥} ∈ Gr_{n−p}(ℂ^{n}).

The complexified tangent bundle of the twistor space F_{l_{1},l_{2},...,l_{m}} splits into vertical and horizontal parts:

V^{ℂ} = ⨁_{i+j even} Hom(V_{i}, V_{j}) ;  H^{ℂ} = ⨁_{i+j odd} Hom(V_{i}, V_{j})  (3)

The bundle V^{ℂ} is the kernel of ζ_{∗}^{ℂ} : T^{ℂ}F_{l_{1},l_{2},...,l_{m}} → T^{ℂ} Gr_{p}(ℂ^{n}), while H^{ℂ} is the complexification of the horizontal distribution H:

H = ⨁_{{i+j odd \atop i<j}} Hom(V_{i}, V_{j}).  (4)

The horizontal distrubuiton is an important tool in the study of minimal surfaces [1]. In general it is not holomorphic, but it contains a holomorphic subdistribution known as the superhorizontal distribution, see [4]:

S = ⨁_{i=1}^{m−1} Hom(V_{i}, V_{i+1}).

Of course H and S coincide precisely when m = 3.

Maps φ : M → F tangent to the horizontal (or superhorizontal) distribution are called horizontal (respectively superhorizontal).

2.2. Harmonic and pluriharmonic maps. The twistor fibration ζ, defined in (2), has applications in the theory of harmonic surfaces since it has the following property (for various versions and levels of generality see [8, 2, 5, 4]):

Let Σ be a Riemann surface. If φ : Σ → F is holomorphic horizontal, then its twistor projection ζ ∘ φ : Σ → Gr_{p}(ℂ^{n}) is a harmonic map.

In particular if M is a Kähler manifold and f : M → F is holomorphic horizontal then the restriction of f to any holomorphic curve in M gives a horizontal holomorphic curve in F and hence a harmonic map into Gr_{p}(ℂ^{n}). Consequently, f is pluriharmonic. To summarize: horizontal holomorphic maps into flag manifolds project (under the twistor projection) to pluriharmonic (in fact (1, 1)-geodesic) maps into complex Grassmannians.

Our aim is to construct pluriharmonic submanifolds of maximal dimension in Gr_{p}(ℂ^{n}). This will be achieved by constructing horizontal submanifolds in a flag manifold fibering over Gr_{p}(ℂ^{n}). In the following Section we will restrict our attention to flag manifolds F_{1,q+1,n}, where p + q = n.

3. The construction

3.1. Flag manifold F_{1,q+1,n}. This flag manifold can be written as {(V_{1}, V_{2}) : V_{1} ⊂ V_{2} ⊂ ℂ^{n}, dim V_{1} = 1, dim V_{2} = q + 1}. Denote by p^{1}, p^{2} the canonical holomorphic fibrations of F_{1,q+1,n} over ℂℙ^{n−1} and Gr_{q+1}(ℂ^{n}) respectively, see Diagram (6). We also have the twistor fibration:

ζ : F_{1,q+1,n} ∋ (V_{1}, V_{2}) → V_{1} ⊕ V_{2}^{⊥} ∈ Gr_{p}(ℂ^{n})  (5)

where p = n − q.

[figure: F_{1,q+1,n}; Gr_{p}(ℂ^{n}); ℂℙ^{n−1}; Gr_{q+1}(ℂ^{n}); ζ; p^{1}; p^{2}]  (6)

It follows from (4) that the horizontal distribution H is spaned by the vertical bundles for p^{1} and p^{2}:

H = ker p_{∗}^{1} ⊕ ker p_{∗}^{2}

The following characterization of the horizontal distribution will prove useful:

Lemma 2. With the notation as above the following holds:

A: As a holomorphic bundle over ℂℙ^{n−1} the flag manifold F_{1,q+1,n} can be identified

with the complex Grassmann bundle Gr_{q} T ℂℙ^{n−1}. B: A holomorphic map ψ : M → F_{1,q+1,n} ≅ Gr_{q} T ℂℙ^{n−1} is horizontal precisely when for each z ∈ M the complex q-space ψ(z) ⊂ T_{ϕ(z)}ℂℙ^{n−1} contains ϕ_{∗}T_{z}M, where ϕ = p^{1} ∘ ψ : M → ℂℙ^{n−1}.

Proof. Consider a flag v = (V_{1}, V_{2}, V_{3}) ∈ F_{1,q+1,n}. Then p^{1}(v) = V_{1} and Hom(V_{1}, V_{2}/V_{1}) ⊂ Hom(V_{1}, V_{3}/V_{1}) = T_{V_{1}}ℂℙ^{n−1} is the complex q-dimensional subspace that corresponds to v.

To prove part (B) we will use the orthogonal description of flags. Take z ∈ M and v = ψ(z) = (V_{1}, V_{2}, V_{3}) ∈ F_{1,q,p−1}. The differential (p_{∗}^{1})_{v} is the obvious projection

T_{v}^{(1,0)}F_{1,q,p−1} = ⨁_{i<j} Hom(V_{i}, V_{j}) → Hom(V_{1}, V_{2} ⊕ V_{3}) = T_{V_{1}}^{(1,0)}ℂℙ^{n−1}

Let A_{z} ∈ T_{z}M. Then ψ_{∗}(A_{z}) is horizontal when it has no component in Hom(V_{1}, V_{3}), i.e. the vector p_{∗}^{1}ψ_{∗}(A_{z}) = φ_{∗}(A_{z}) lies in Hom(V_{1}, V_{2}). But by the identification (A) ψ(z) = Hom(V_{1}, V_{2}) is a q-dimensional subspace of T_{V_{1}}ℂℙ^{n−1} = Hom(V_{1}, V_{2} ⊕ V_{3}). It follows that φ_{∗}(A_{z}) ∈ ψ(z) precisely when ψ_{∗}(A_{z}) is horizontal. □

Let us consider now a horizontal immersion ψ : M → F_{1,q+1,n} and let φ = p^{1} ∘ ψ. Suppose further that φ has constant rank, so that Σ = φ(M) is a k-dimensional submanifold of ℂℙ^{n−1}. For z ∈ M, y = φ(z) the space ϕ_{∗}(T_{z}M) = T_{y}Σ is contained in ψ(z), and the quotient ψ(z)/T_{y}Σ is a (q − k)-dimensional subspace of the normal bundle ν_{Σ} = T ℂℙ^{n−1}/T Σ. We can thus view M as a submanifold of the Grassman bundle Gr_{q−k}ν_{Σ}.

One can invert this procedure: consider a k-dimensional complex immersion α : Σ → ℂℙ^{n−1}. There is a 1-1 correspondence between (q − k)-subspaces p_{z} in (ν_{Σ})_{z} and qdimensional subspaces p̂_{z} in T_{y}ℂℙ^{n−1} containing α_{∗}(T_{z}Σ), where y = α(z). This gives a holomorphic immersion

α̃ : Gr_{q−k} ν_{Σ} → Gr_{q} T ℂℙ^{n−1} ≅ F_{1,q+1,n},

where

α̃(p_{z}) = α_{∗}(p̂_{z}).  (7)

Moreover, it follows from Lemma 2 that the map α̃ is horizontal. We can now obtain pluriharmonic immersions by composing α̃ with the twistor projection ζ.

3.2. The dimensions. Let us calculate the (complex) dimension of Gr_{q−k} ν_{Σ}, assuming that α : Σ → ℂℙ^{n−1} is an immersion, and dim Σ = k. We have rank ν_{Σ} = n − 1 − k and then

dim Gr_{q−k} ν_{Σ} = (q − k)(n − 1 − k − (q − k)) + k
= (q − k)(p − 1) + k

We want the dimension of the pluriharmonic submanifold maximal, i.e.

(q − k)(p − 1) + k = (p − 1)(q − 1) + 1.

This implies that p = 2 or k = 1. As we do not want restrictions on the space Gr_{p}(ℂ^{n}), we opt for the case k = 1, in other words, we will assume from now on that Σ is a Riemann surface. Since Gr_{q−k} ν_{Σ} is a horizontal submanifold in F_{1,q+1,n}, it is transversal to the fibres of ζ : F_{1,q+1,n} → Gr_{p}(ℂ^{n}). Consequently, the projection of Gr_{q−1} ν_{Σ} to the base gives a pluriharmonic immersion of dimenision (p − 1)(q − 1) + 1. The results of this discussion are summarised in the following theorem.

Theorem 3. If α : Σ → ℂℙ^{n−1} is a nonsingular holomorphic curve then the map α̃, defined in (7), followed by the twistor projection ζ : F_{1,q+1,n} → Gr_{p}(ℂ^{n}) gives a (p − 1)(q − 1) + 1dimensional pluriharmonic immersion of Gr_{q−1} ν_{Σ} in Gr_{p}(ℂ^{n}). Moreover, if α is linearily full then this immersion is not ±holomorphic.

A curve in ℂℙ^{n−1} is linearily full if it is not contained in a proper projective subspace of ℂℙ^{n−1}.

Proof. In order to prove the theorem it remains to check that the map ζ ∘ α̃ is not ±holomorphic. We will show that ζ ∘ α̃ is not holomorphic by constructing a holomorphic curve in Gr_{q−1} ν_{Σ} which has a nonholomorphic projection to Gr_{p}(ℂ^{n}). Recall that for v = (V_{1}, V_{2}, V_{3}) ∈ F_{1,q+1,n} = F_{1,q,p−1} we have

T_{v}^{1,0}F_{1,q,p−1} = ⨁_{i<j} Hom(V_{i}, V_{j}).

Moreover, ζ(v) = (V_{1} ⊕ V_{3}, V_{2}) = (W_{1}, W_{2}) ∈ F_{p,q} = Gr_{p}(ℂ^{n}) and

T_{ζ(v)}^{1,0} Gr_{p}(ℂ^{n}) = Hom(W_{1}, W_{2}) = Hom(V_{1} ⊕ V_{3}, V_{2}).

The complexified differential ζ_{∗} : T_{v}^{ℂ}F → T_{ζ(v)}^{ℂ} Gr_{p}(ℂ^{n}) is the obvious projection

⨁_{i≠j} Hom(V_{i}, V_{j}) → Hom(W_{1}, W_{2}) ⊕ Hom(W_{2}, W_{1})  (8)

Choose a local coordinate z on Σ and write the map α : Σ → ℂℙ^{n−1} as a projectivisation of a vector function a : ℂ → ℂ^{n}. Since α is linearily full, the vectors a(z), a′(z), …, a^{(n−1)}(z) are linearily independent in ℂ^{n}. Define the following subspaces of ℂ^{n}:

V_{1}(z) = ℂa(z)
V_{2}(z) = ℂa(z) ∧ a′(z) ∧ a′′(z) ∧ … ∧ a^{(q)}(z).

Then

χ : Σ ∋ z ↦ (V_{1}(z), V_{2}(z), ℂ^{n}) ∈ F_{1,q+1,n}

is a holomorphic curve in the flag manifold F_{1,q+1,n}. Let us denote by (V_{1}(z), V_{2}(z), V_{3}(z)) the corresponding orthogonal representation of the curve, so

V_{1} = V_{1}, V_{2} = V_{1} ⊕ V_{2}, and ℂ^{n} = V_{1} ⊕ V_{2} ⊕ V_{3}.

Since f^{(q+1)} = \frac{∂}{∂z}f^{(q)} does not belong to V_{2}(z), it follows from equation (1) that χ_{∗}(\frac{∂}{∂z}) has a nonzero component in Hom(V_{2}, V_{3}) and this gives a nonzero component of ζ_{∗} ∘ χ_{∗}(\frac{∂}{∂z}) in

T^{0,1} Gr_{p}(ℂ^{n}) = Hom(W_{2}, W_{1}) = Hom(V_{2}, V_{1} ⊕ V_{3}).

It follows that ζ ∘ α̃ : Gr_{q−1} ν_{Σ} → Gr_{p}(ℂ^{n}) is not holomorphic. Similarly, a(z) and a′(z) are linearily independent so ζ_{∗} ∘χ_{∗}(\frac{∂}{∂z}) has a nonzero component in Hom(V_{1}, V_{2}) and hence in T^{(1,0)} Gr_{p}(ℂ^{n}), so the map ζ ∘ α̃ is not antiholomorphic. □

Note that it is sufficient to assume in the theorem that Σ is not contained in a qdimensional projective subspace of ℂℙ^{n−1}. Also, V_{3} must be nontrivial, so we need q+1 < n, but if q = n − 1 then Gr_{q}(ℂ^{n}) is (antiholomorphically) isometric to ℂℙ^{n−1}, and the only nontrivial pluriharmonic submanifolds in ℂℙ^{n−1} are harmonic surfaces.

Let us have a look a the situation when Σ ⊂ ℂℙ^{n−1} is far from being linearily full.

Example 4. Write ℂ^{n} = ℂ^{2} ⊕ ℂ^{n−2} as an orthogonal direct sum and consider the immersion φ of the manifold M = Gr_{1}(ℂ^{2}) × Gr_{q−1}(ℂ^{n−2}) into Gr_{p}(ℂ^{n}):

φ : Gr_{1}(ℂ^{2}) × Gr_{q−1}(ℂ^{n−2}) ∋ (V_{1}, V_{2}) → V_{1} ⊕ V_{2}^{⊥} ∈ Gr_{p}(ℂ^{n})

Note that dim M = 1+(q −1)((n−2)−(q −1)) = (p−1)(q −1)+1. Moreover, φ is neither holomorphic nor antiholomorphic since it is holomorphic in the first, and antiholomorphic in the second variable. However, the Hom(V_{2}, V_{3}) component vanishes (it can be calculated as in the proof of Theorem 3). So even though φ is not ±holomorphic, it is not interesting, as its image is a holomorphic submanifold of Gr_{p}(ℂ^{n}). Of course Gr_{1}(ℂ^{2}) ≅ ℂℙ^{1} and M is really Gr_{q−1} ν_{ℂℙ}^{1}, with the holomorphic structure along the fibres reversed; in fact map φ factors through the holomorphic map

Gr_{1}(ℂ^{2}) × Gr_{p−1}(ℂ^{n−2}) ∋ (V_{1}, V_{2}) → V_{1} ⊕ V_{2} ∈ Gr_{p}(ℂ^{n}).

4. Superhorizontal maps

How general is the construction described in Section 3? We pointed out in 3.1 that if M is a holomorphic submanifold of F_{1,q+1,n} and its projection Σ to the base has constant rank k then M is a submanifold of Gr_{q−k} ν_{Σ}. We want M to have maximal dimension so Gr_{q−k} ν_{Σ} is preferred over its submanifolds. But is it possible to construct nontrivial pluriharmonic submanifolds of maximal dimension starting from horizontal holomorphic submanifolds in other flag manifolds? It turns out that we are forced to choose F_{1,q+1,n}:

Suppose that we have a horizontal immersion ψ : M → F_{p_{1},p_{2},p_{3}}, holomorphic with respect to the twistor fibration ζ : F_{p_{1},p_{2},p_{3}} → Gr_{p}(ℂ^{n}), where p = p_{1} + p_{3}, q = p_{2}. Assume further that ψ composed with the twistor projection is not ±holomorphic. Take z ∈ M, and denote ψ(z) = (V_{1}, V_{2}, V_{3}), A = ψ_{∗}(T_{ψ(z)}^{1,0}M) ⊂ T_{ψ(z)}F_{p_{1},p_{2},p_{3}}. Then A is a subspace of the horizontal space

H_{ψ(z)} = Hom(V_{1}, V_{2}) ⊕ Hom(V_{2}, V_{3}).

For u, w ∈ A the vector [u, w] cannot have vertical components, so [w, u] = 0 and treating

T_{v}^{1,0}F_{p_{1},p_{2},p_{3}} = ⨁_{i<j} Hom(V_{i}, V_{j})

as a matrix subalgebra of 𝔤𝔩(n, ℂ) (note that it is nilpotent) we find that A ⊂ T_{v}^{1,0}F is an abelian algebra (this is a special case of a more general statement, cf. [15] or the discussion in [7]). Moreover, since ψ is not ±holomorphic it follows from (8) that A cannot lie in Hom(V_{1}, V_{2}) nor in Hom(V_{2}, V_{3}). Note that if A is an abelian algebra satisfying these conditions then the map A ∋ x ↦ exp(x) · v ∈ F is holomorphic and horizontal. The search of the maximal dimension of pluriharmonic maps thus reduces to algebra:

Lemma 5. Consider a flag (V_{1}, V_{2}, V_{3}) ∈ F_{p_{1},p_{2},p_{3}}, p_{2} ≥ 2, and let A ⊂ Hom(V_{1}, V_{2}) ⊕ Hom(V_{2}, V_{3}) be an abelian matrix algebra of maximal dimension. If A is contained neither in Hom(V_{1}, V_{2}) nor in Hom(V_{2}, V_{3}) then

(i) dim A = (p − 1)(q − 1) + 1, where p = p_{1} + p_{3}, q = p_{2}, moreover, (ii) if q ≥ 3 then p_{1} = 1 or p_{3} = 1.

Proof. The abelian Lie algebra that arises in the construction in Part 3 is Hom(V_{1}, U) ⊕ Hom(W, V_{3}), where (V_{1}, V_{2}, V_{3}) ∈ F_{1,q,p−1} and V_{2} = U ⊕ W, dim U = 1. This impliest that

dimA ≥ (p − 1)(q − 1) + 1  (9)

The proof is by induction on n. Statements (i) and (ii) are obvious for small n.

We can apply the matrix transpose to A so, without loss of generality, we may assume that p_{1} ≥ p_{3}. Suppose first that

dim A ≤ dim Hom(V_{2}, V_{3}).

From this and inequality (9) whe have

(p − 1)(q − 1) + 1 ≤ p_{3}q

and, since p = p_{1} + p_{3}, p_{1} ≥ p_{3}, we get

p_{3}(q − 1) ≤ p_{1}(q − 1) ≤ q − 2 + p_{3}

and then

(p_{3} − 1)(q − 2) ≤ 0.

It follows that p_{3} = 1 so (ii) holds, and then so does (i) for q > 2. We leave the case q = 2 for the time being and consider now the case

dim A > dim Hom(V_{2}, V_{3}).

This means that there exists a nonzero element h ∈ A with no component in Hom(V_{2}, V_{3}). Denote U = h(V_{1}) ⊂ V_{2} and choose a subspace W in V_{2} so that V_{2} = U ⊕ W. Any element with nonzero component in Hom(U, V_{3}) has nontrivial commutator with h, so

A ⊂ Hom(V_{1}, W) ⊕ Hom(W, V_{3}) ⊕ Hom(V_{1}, U)

Denote q_{1} = dim U, q_{2} = dim W, and let A′ be the orthogonal projection of A to Hom(V_{1}, W) ⊕ Hom(W, V_{3}). Then A ⊂ A′ + Hom(V_{1}, U), and

dim A ≤ dim A′ + dim Hom(V_{1}, U).

We have one of the following three cases:

(1) A′ ⊂ Hom(V_{1}, W) — this is impossible since then A ⊂ Hom(V_{1}, V_{2}),

(2) A′ ⊂ Hom(W, V_{3}) — then A ⊂ Hom(V_{1}, U) ⊕ Hom(W, V_{3}), and

dim A ≤ p_{1}q_{1} + q_{2}p_{3}

but p_{1} + p_{3} = p and q_{1} + q_{2} = q so the maximal value for p_{1}q_{1} + q_{2}p_{3} is equal to (p − 1)(q − 1) + 1, and is attained if p_{3} = 1 (provided that q > 2), so (i) and (ii) is true. If q = 2 then dim U = dim W = 1 and Hom(V_{1}, U) ⊕ Hom(W, V_{3}) is an abelian algebra of dimension p_{1} + p_{3} = p = (p − 1)(q − 1) + 1, so (i) holds.

(3) If neither (1) nor (2) is satisfied then, since A′ is abelian, we have by induction: dim A′ ≤ (p − 1)(q_{2} − 1) + 1. Then

(p − 1)(q − 1) + 1 ≤ dim A ≤ (p − 1)(q_{2} − 1) + 1 + p_{1}q_{1}

which again implies that p_{3} = 1. What remains is the case (i) when dim A ≤ dim Hom(V_{2}, V_{3}), q = 2. This gives qp_{3} ≥ p,

so 2p_{3} ≥ p_{1} + p_{3} and hence p_{1} = p_{3}. Write elements h ∈ A as (h_{1}, h_{2}) where h_{1} ∈ Hom(V_{1}, V_{2}) and h_{2} ∈ Hom(V_{2}, V_{3}). Then for h′ = (h′_{1}, h′_{2}) ∈ A we have

[h, h′] = 0 if and only if h′_{2}h_{1} = h_{2}h′_{1}.

Since p_{1} = p_{3}, without loss of generality we may assume that rank h_{1} ≥ rank h_{2}. The idea is to use nonzero h = (h_{1}, h_{2}) ∈ A with rank of h_{2} as low as possible. Assume first that there exists h ∈ A with rank h_{2} < 2 and consider h′ = (h′_{1}, h′_{2}) ∈ A. If [h, h′] = 0, h ≠ 0, then h′_{2}(im h_{1}) ⊂ im h_{2}, so

codim π^{2,3}A ≥ dim(im h_{1})(p_{3} − dim im h_{2}) ≥ p_{3} − 1

where π^{2,3} denotes the canonical projection to Hom(V_{2}, V_{3}). Then

dim π^{2,3}A ≤ 2p_{3} − p_{3} + 1 = p_{3} + 1.

Now if dim A > dim π^{2,3}A then there exists an element h in A with no component in Hom(V_{2}, V_{3}). Then define U = h(V_{1}) and proceed as before. Otherwise dim A = dim π^{2,3}A ≤ p_{3} + 1, so p_{1} ≤ 1 by (9).

Now let us assume that all we can find is h ∈ A such that rank h_{1} = rank h_{2} = 2. Then ker h_{2} = 0 so [h, h′] = 0 implies that ker h_{1} ⊂ ker h′_{1}. Also, im h_{1} = V_{2}, so im h′_{2} ⊂ im h_{2}. Choose V′_{1} so that V_{1} = ker h_{1} ⊕ V′_{1}. It follows that A can be identified with a subalgebra of Hom(V′_{1}, V_{2}) ⊕ Hom(V_{2}, im h_{2}) and hence its dimension can’t exceed (p′ − 1)(q − 1) + 1 (by induction; p′ = dim V′_{1} + dim im h_{2}). So (9) implies that im h_{2} = V_{3} and ker h_{1} = 0, i.e. V_{1} and V_{3} are 2-dimensional.

Let us find the space H of all matrices h′ ∈ Hom(V_{1}, V_{2}) ⊕ Hom(V_{2}, V_{3}) that commute with h. By choosing suitable bases in V_{i} and writing h and h′ in block matrix form we get

h = ( \begin{matrix}0 & 0 & 0 \\ 1_{2} & 0 & 0 \\ 0 & 1_{2} & 0 \end{matrix} ) and h′ = ( \begin{matrix}0 & 0 & 0 \\ a & 0 & 0 \\ 0 & a & 0 \end{matrix} )

where 1_{2} denotes the 2 × 2 unit matrix and a is an arbitrary 2 × 2 matrix. This shows that dim H = 4. But we have dim A ≥ 4 and A ⊂ H so A = H which is not possible since H is not a Lie algebra. □

Remark 6. Note that if q = 1 then V_{2} ≅ ℂ and we can write H_{v} as V_{1}^{∗} ⊕ V_{3} and the Lie bracket restricted to Λ^{2}H_{v} is given by the formula

[(α, v), (α′, v′)] = α ⊗ v′ − α′ ⊗ v.

This is zero only if α = λα′, v = λv′, so (α, v) and (α′, v′) are linearily dependent and then dim A = 1 (so (i) in Lemma 5 is true but it is possible that p_{1} > 1 or p_{3} > 1.

The case q = 1 (or q = n − 1) is not interesting since this means constructing pluriharmonic maps into ℂℙ^{n−1}. They must have rank 1 so they are harmonic and they may arise from F_{k,1,n−k−1} for any value of k (see [4]).

So far we considered only flag manifolds of the type F = F_{ℓ_{1},ℓ_{2},...,ℓ_{m}} where m = 3. For larger values of m the horizontal distribution H no longer coincides with the superhorizontal distribution S and is not holomorphic. So it makes sense to look for submanifolds in F which are superhorizontal. But it turns out that in general the superhorizontal distribution has to large codimension to accommodate superhorizontal submanifolds of high dimension.

To see this suppose that M is a superhorizontal holomorphic submanifold in F = F_{ℓ_{1},ℓ_{2},...,ℓ_{m}} of dimension (p−1)(q −1)+1, where p = ℓ_{1} +ℓ_{3} +…, q = ℓ_{2} +ℓ_{4} +…, p+q = n and m ≥ 4. If we look for a nontrivial abelian subalgebra of dimension (p − 1)(q − 1) + 1 in T_{v}F, we have

(p − 1)(q − 1) + 1 = (−1 + p_{1} + p_{3} + …)(−1 + p_{2} + p_{4} + …) + 1 > p_{2}p_{3},

so dim A > dim Hom(V_{2}, V_{3}), and then we can repeat the argument as in the proof of Proposition 5. Consequently, we must have F = F_{ℓ_{1},ℓ_{2},...,ℓ_{m}} with m = 3 and then F = F_{1,q,p−1}. (The case F = F_{p−1,q,1} does not give anything new as this manifold is antiholomorphically isometric to F_{1,q,p−1}.)

5. Isotropic pluriharmonic maps

Eschenburg and Tribuzy consider the class of isotropic pluriharmonic maps, defined in terms of associated families (see [10] for details). They show that in inner symmetric spaces such maps (subject to being full, i.e. not contained in a proper totally geodesic subspace) must arise from a twistorial fibration of a flag manifold (or a flag domain) over the symmetric space. Moreover, they show that in the compact case such maps are in fact (1,1)-geodesic immersions.

We can thus construct all full (1,1)-geodesic immersions of maximal rank in complex Grassmannians, using the construction in Section 3.

Theorem 7. Let φ : M → Gr_{p}(ℂ^{n}) be a full isotropic (1,1)-geodesic immersion (M is connected Kähler) which is not ±holomorphic and has maximal rank. If 2 < p < n−2 then φ arises as the twistor projection ζ of α̃ : Gr_{s−1} ν_{Σ} → F_{1,s+1,n}, where s = p or s = n − p, α : Σ → ℂℙ^{n−1} is a ±holomorphic curve, and α̃ is defined by Equation 7.

Remark 8. The cases p = 2 and p = n − 2 are exceptional:

(i) There is a family of (1,1)-geodesic maps for various values of k (see Section 3.2), where k is the rank of the map α : Σ → ℂℙ^{n−1}.

(ii) In particular if k = n−2 one gets the construction of pluriharmonic maps described in [9, 11]. The idea is to start with a hypersurface Σ in ℂℙ^{n−1}; the map

Σ ∋ l → (l, T_{l}Σ) ∈ Gr_{n−2} T ℂℙ^{n−1} ≅ F_{1,n−1,n}

is then a horizontal holomorphic map to the twistor space and projects to a pluriharmonic map Σ → Gr_{2}(ℂ^{n}). This is the same as the map (7) since q = n − 2 = k so Gr_{q−k} ν_{Σ} can be trivially identified with Σ.

(iii) Finally, note that the pluriharmonic maps into Gr_{2}(ℂ^{n}) mentioned in (i) arise from the flag manifold F_{1,2,n−3}; there are also maps that arise from F_{1,n−2,1} (here k = 1 and p = 1 so they are in the same flavour as the maps in Theorem 7. But the proof of Lemma 5 does not exclude the possibility of maps arising from F_{p,2,n−p−2} with p > 1 (cf. item (2) in the proof).

Acknowledgements. The author wishes to thank Jost Eschenburg for encouragement and valuable conversations. This research was partially supported by KBN in Poland and EDGE (Odense node).

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Instytut Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, 30–059 Kraków, Poland

E-mail address: kobak@im.uj.edu.pl