Pieri-Type Intersection Formulas and Primary Obstructions for Decomposing 2-forms

Sinan Sertöz (Ankara)

Appeared in Colloquium Mathematicum, 87, (2001), 201-210.

Abstract. We study the homological intersection behaviour for the Chern cells of the universal bundle of G(d, Q_{n}), the space of [d]-planes in the smooth quadric Q_{n} in ℙ^{n+1} over the field of complex numbers. For this purpose we define some auxiliary cells in terms of which the intersection properties of the Chern cells can be described. This is then applied to obtain some new necessary conditions for the global decomposability of a 2-form of constant rank.

1. Introduction. In this article we study from a purely projective geometric point of view the obstructions to globally decomposing a 2-form. It was shown by Dibağ that the vanishing of certain Chern classes is necessary for such a decomposition, [3]. We construct new classes whose nonvanishing implies the nonvanishing of the Chern classes. Moreover some vanishing patterns of these new classes imply the vanishing of the Chern class obstructions. This is achieved by studying the intersection structure of the integral homology generated by the Chern cells. Our methods are purely geometric and reveal the required products up to a nonzero multiplicative constant. However this suffices for our purposes since we eventually check for vanishing of obstructions. In the case of maximal planes these coefficients can be explicitly calculated. This is done by Hiller and Boe in [7] where the authors consider the case of type B maximal isotropic Grassmannians. Type D (which is a consequence of the result in type B) appeared in [9]. The results of [7] are further reproved by Pragacz and Ratajski in [11] by using divided differences. Recently similar calculations in type B are done by Sottile, [14].

AMS Subject classification: 14M15, (14M17, 14N10)

In the non-maximal case these calculations are due to Pragacz and Ratajski, see [12]. However to adopt these general formulas for our cases would lead to complicated combinatorial formulas. By checking only nonvanishing conditions we are able here to present a purely geometrical argument which suffices for our results.

We denote by G(d, Q_{n}) the space of complex projective [d]-planes lying in the smooth quadric hypersurface Q_{n} of ℙ^{n+1}. Dibağ has shown that G(d, Q_{n}) represents A_{d+1}^{(n+2)}, the space of normalized 2-forms in ℝ^{n+2} of rank 2(d + 1), on which the Stiefel bundle V_{n+2,2(d+1)} of orthonormal 2(d + 1)- frames in ℝ^{n+2} induces a principal U(d + 1)-bundle, see [2, 3].

In general if ω is a 2-form of constant rank 2(d + 1) on a trivial ℝ^{n+2} bundle E over some base space B, then it can be represented by a map ω_{1} : B → A_{d+1}^{(n+2)}. Lifting this map to V_{n+2,2(d+1)} is equivalent to decomposing the 2-form ω globally as ω = y_{1}∧y_{d+2}+⋯+y_{d+1}∧y_{2(d+1)} for some 1-forms on E. Then the images ω_{1}^{∗}(c_{i}) ∈ H^{2i}(B; ℤ) of the Chern classes c_{i} ∈ H^{2i}(A_{d+1}^{(n+2)}; ℤ), i = 0, ..., d+1, of the principal U(d+1)-bundle V_{n+2,2(d+1)} necessarily vanish. If E is not trivial then the above geometry is analyzed on a certain subbundle S_{ω} of E, depending on ω, and its triviality is another necessary condition for the decomposability of ω, [3].

We define some cohomology classes P D Ω_{i} ∈ H^{ℓ−i}(A_{d+1}^{(n+2)}; ℤ), where ℓ = dim A_{d+1}^{(n+2)} and i = 0, …, d + 1, and show that if ω_{1}^{∗}(P D Ω_{s}) = 0 for some 0 ≤ s ≤ d + 1, and ω_{1}^{∗}(P D Ω_{s+i}) ≠ 0 for i = 1, …, d + 1 − s, then ω_{1}^{∗}(c_{i}) = 0 for i = 1, …, d + 1 − s. Moreover if ω_{1}^{∗}(P D Ω_{s}) ≠ 0 for some s, then ω_{1}^{∗}(c_{i}) ≠ 0 for all i = 1, …, d + 1 − s, see Theorem 3 and Corollary 4. When n = 2d, a trivial line bundle splits off the universal bundle on each irreducible component, V_{0} and V_{1}, of G(d, Q_{n}) forcing c_{d+1} to vanish. In this case s > 0 if it exists. These results occupy the last section after we establish in Section 3 the intersection properties of Chern cells.

For background on intersection problems we refer to [1, 5, 6]. For recent applications one can refer to [10, 12, 13, 14]. For the existence and the decomposability of 2-forms see [2, 3, 8]. And finally for 2-forms on spheres see [2, 4]

It is my pleasure to thank İ. Dibağ for his constant support and encouragement, and for saving me from an error in an early version of this manuscript.

2. Preliminaries. In this section we summarize some constructions which help us to understand the geometry of the space of d-planes lying in a smooth quadric Q_{n}. First we define a set of points which we propose to call the skeleton points and use in describing flags and Schubert cells. We refer to [13, pp 203-207] for further details and here briefly describe the main lines for completeness.

For two points p = (p_{1}, …, p_{N}) and q = (q_{1}, …, q_{N}) in ℂ^{N} we say that p and q are f-orthogonal if p·q = p_{1}q_{1} +⋯+p_{N}q_{N} = 0, and m-orthogonal if p·\overline{q} = p_{1}\overline{q_{1}}+⋯+p_{N} \overline{q_{N}} = 0, where the overbar denotes complex conjugation. When p and q are used as homogeneous coordinates of the corresponding points in the projective space the same terminology prevails.

Assume n = 2m. The skeleton points of Q_{2m} is a set of 2m + 2 points in ℙ^{2m+1} chosen as follows:

(i) choose p_{0} in Q_{2m} arbitrarily. This means that p is f-orthogonal to itself. (ii) Once p_{0}, ..., p_{k−1} are chosen, where 1 ≤ k ≤ m, choose p_{k} as any point of Q_{2m} which is both f-orthogonal and m-orthogonal to the join p_{0}⋁ ⋯ ⋁p_{k} but not in the join itself. The set of points in ℙ^{2m+1} satisfying these conditions is a 2(m − k) dimensional subspace so a choice is always possible. (iii) After having chosen p_{0}, ..., p_{m} the remaining m + 1 points are chosen as the complex conjugates of these where indexing is done as follows:

p_{m+i} = \overline{p_{m+1−i}}, i = 1, ..., m + 1.

Here again the overbar denotes complex conjugation.

These points p_{0}, ..., p_{2m+1} all lie in Q_{2m} and their join is the whole space ℙ^{2m+1}. The particular way we choose and index them enables us to build a link between geometry and algebra. This can be seen in the following construction.

For any subset L of the set I_{2m+1} = {0, 1, ..., 2m + 1} define S_{L} as the intersection of Q_{2m} with the join of those skeleton points whose index is in L;

S_{L} = Q_{n} ⋂ ( ⋁_{j∈L} p_{j}) .

The link between geometry and algebra comes into play at this stage; the dimension of this space S_{L} is determined by the indexing set L. For this define a particular subset of L which affects the dimension of S_{L}.

J(L) = {i ∈ I_{m} | i ∈ L and 2m + 1 − i ∈ L }.

In other words J(L) contains the indices of only those skeleton points among p_{0}, ..., p_{m} whose complex conjugates also lie in S_{L}.

In [13, Lemma 1.3] we proved that

dim_{ℂ} S_{L} = \{ \begin{matrix}#L − 2 & if J(L) ≠ ∅ \\ #L − 1 & if J(L) = ∅. \end{matrix}

Now we are ready to construct two dual flags for Q_{2m} . The first one is called the A-flag and consists of a nested sequence of subvarieties of Q_{2m}

A_{0} ⊆ ⋯ ⊆ A_{m−1} ⊆ A_{m_{0}}, A_{m_{1}} ⊆ A_{m+1} ⊆ ⋯ ⊆ A_{2m} = Q_{2m}

such that A_{i} − A_{i−1} is an open cell of dimension i, and each A_{i} is chosen as follows

(i) A_{i} = S_{{0,1,...,i}} for i = 0, ..., m − 1 (ii) A_{m_{0}} = S_{{0,1,...,m}} and A_{m_{1}} = S_{{0,1,...,m−1,m+1}} (iii) A_{m+i} = S_{{0,1,...,m+1+i}} for i = 1, ..., m. Note that dim A_{i} = i for i = 0, …, 2m, where the indices m_{0} and m_{1} are considered to be different as indices but both equal to m as values.

The second flag is the dual flag for the A-flag and is called the B-flag. It also consists of a nested sequence of subvarieties of Q_{2m}

B_{0} ⊆ ⋯ ⊆ B_{m−1} ⊆ B_{m_{0}}, B_{m_{1}} ⊆ B_{m+1} ⊆ ⋯ ⊆ B_{2m} = Q_{2m}

where each B_{i} is chosen as follows

(i) B_{i} = S_{{2m+1,2m,...,2m+1−i}} for i = 0, ..., m − 1 (ii) B_{m_{0}} = S_{{2m+1,2m,...,m+2,m}} and B_{m_{1}} = S_{{2m+1,2m,...,m+1}} if m is even,

B_{m_{1}} = S_{{2m+1,2m,...,m+2,m}} and B_{m_{0}} = S_{{2m+1,2m,...,m+1}} if m is odd (iii) B_{m+i} = S_{{2m+1,2m,...,m−i}} for i = 1, ..., m. Note again that dim B_{i} = i for i = 0, …, 2m.

For the corresponding constructions in the n = 2m + 1 case we refer the reader to [13, pp 205-207].

To conclude this section we summarize the construction of Schubert cells on G(d, Q_{2m} ), the space of d-planes in Q_{2m}. In the notation of [3] G(d, Q_{2m} ) is A_{d+1}^{(n+2)}. A Schubert symbol for G(d, Q_{2m} ) is a finite sequence of integers a = (a_{0}, ..., a_{d}), d ≤ m, satisfying the conditions

(i) 0 ≤ a_{0} < ⋯ < a_{d} ≤ 2m (ii) a_{i} + a_{j} ≠ 2m for i < j. This conditions is to avoid assigning different Schubert symbols to the same cell. See [3, p 506]. Here again m_{0} and m_{1} are used as two different entities but both having the value m, so if one of them appears in the sequence a the other does not according to (ii).

The Schubert cell corresponding to the Schubert symbol a is a subvariety of G(d, Q_{2m} ) defined as

Ω_{a_{0}···a_{d}} = {P ∈ G(d, Q_{2m} ) | dimℂ(P ∩ A_{a_{i}}) ≥ i }.

Here A_{a_{i}} denotes the corresponding member of the A-flag. It turns out that

dim_{ℂ} Ω_{a_{0}···a_{d}} = a_{0} + ⋯ + a_{d} − d(d + 1) + e

where

e = #{(a_{i}, a_{j}) | i < j and a_{i} + a_{j} < n }.

For further details on the intersection properties of these cells we refer to [3, 13].

3. Intersecting Chern Cells. Following Dibağ the i-th Chern cell Ω_{i} of the principal U(d + 1)-bundle V_{n+2,d+1} (A_{d+1}^{(2n)}; U(d + 1)) is defined in terms of Schubert cells on G(d, Q_{n}) as

Ω_{i} = Ω_{0···_{(d−i+1)}···d+1}, 0 ≤ i ≤ d + 1

when n ≥ 2d + 3. (Here ˆ (d − i + 1) means “omit d − i + 1”). The restriction on n ensures that the condition a_{i} + a_{j} ≠ n for i < j holds in the Schubert symbols corresponding to the Chern cells, which is important to avoid redundant representations.

It turns out that the homology duals of the Chern cells play a crucial part in the intersection behaviour of the Chern classes. These are defined as follows;

Δ_{j} = Ω_{n−d−1···_{(n−d−1+j)}···n}, 0 ≤ j ≤ d + 1.

Note that Δ_{j} is the ‘dual’ of Ω_{j}, i.e Ω_{j}^{t} = Δ_{j} in the notation of [3]. A direct calculation shows that dim_{ℂ} Ω_{i} = codim_{ℂ}Δ_{i} = i, for 0 ≤ i ≤ d + 1.

The intersection properties of the Chern cells can now be described fully in terms of the dual cells: the i-th Chern cell non trivially intersects a cell if and only if this cell is the j-th dual Chern cell with j not bigger than i, and in that case the intersection is precisely a multiple of the (i − j)-th Chern cell. We can now formulate this in the following theorem;

Theorem 1. Let Ω_{i} and Δ_{j} be as defined above and let Ω be any Schubert cell of G(d, Q_{n}) with n ≥ 2d + 3. Then

Ω_{i} · Ω ≠ 0 if and only if Ω = Δ_{j} with 0 ≤ j ≤ i ≤ d + 1.

And moreover in that case we have

Ω_{i} · Δ_{j} = αΩ_{i−j}, 0 ≤ j ≤ i ≤ d + 1,

where α is a nonzero integer.

Remark. P. Pragacz has communicated these coefficients as powers of 2. In fact Hiller and Boe have shown in [7] that for the maximal plane case, i.e. the n = 2d case, these coefficients are indeed powers of 2, see also [11, 14]. We will deal with the 2d ≤ n < 2d + 3 cases in the next section. However we are only interested in the obstruction theoretical properties of these intersections so it only matters for us if the coefficients are zero or not. By appealing to some general facts about Schubert cycles and Bruhat order it is possible to give a shorter proof of this theorem but we prefer this approach which is elementary and exhibits the inner workings of geometry.

Proof of Theorem 1. We will give the proof for the n = 2m case which reflects the main geometric ideas involved. The n = 2m + 1 case is similar and is omitted. First we note that Ω_{0} is a point and hence non trivially intersects only Δ_{0}, with α = 1. Next let 1 ≤ i ≤ d: Suppose Ω_{i} intersects non trivially a Schubert cell Ω whose Schubert symbol is a = (a_{0}, ..., a_{d}). Let P be a d-plane in G(d, Q_{n}) which lies in the intersection Ω_{i} · Ω. Then P must satisfy simultaneously the Schubert conditions dictated by the two symbols (0, ..., ˆ (d − i + 1), ..., d + 1) and (a_{0}, ..., a_{d}) of Ω_{i} and Ω respectively. Using the A-flag of G(d, Q_{n}) the first symbol (0, ..., ˆ (d − i + 1), ..., d + 1) implies that the d-plane P contains the join p_{0}⋁ ⋯ ⋁p_{d−i} and itself lies in the join p_{0}⋁ ⋯ ⋁p_{d+1}. Here we used the description of the spaces A_{i} of the A-flag. Next we use the dual B-flag to interpret the second symbol. The a_{0} of a now requires that P intersects the space B_{a_{0}}, but dim_{ℂ} B_{a_{0}} = a_{0}.

The d-dimensional plane q lies in the d + 1 dimensional join p_{0}⋁ ⋯ ⋁p_{d+1}. Then this join must have at least a point common with B_{a_{0}}, which forces (d + 1) + a_{0} ≥ n or equivalently a_{0} ≥ n − d − 1. Combining this with the general properties of Schubert symbols we have

n − d − 1 ≤ a_{0} < ⋯ < a_{d} ≤ n.

This means that we have to choose d + 1 integers from the interval [n − d − 1, n]. But there are already only d + 2 integers in this interval so we choose all the integers from this interval except one;

(a_{0}, ..., a_{d}) = (n − d − 1, ..., ˆ (n − d − 1 + j), ..., n), 0 ≤ j ≤ d + 1.

This is precisely the definition of Δ_{j} and thus the first part of the theorem is proved. That j must not exceed i will follow from the proof of the second part of the theorem.

To prove the second part of the theorem we assume that the intersection of Ω_{i} with Δ_{j} is nonempty. We may again assume that i > 0. Assume that P is a d-plane lying in the nonempty intersection Ω_{i} ·Δ_{j}. We know from the above analysis that P must contain the join p_{0}⋁ ⋯ ⋁p_{d−i} and must lie in the d + 1 dimensional space defined by the join p_{0}⋁ ⋯ ⋁p_{d+1}. These conditions are imposed on P because it belongs to Ω_{i}. Now we inspect what further conditions will be imposed on P by forcing it to belong to Δ_{j} as well.

The Schubert symbol of Δ_{j} is

(a_{0}, ..., a_{d}) = (n − d − 1, ..., ˆ (n − d − 1 + j), ..., n) 1 ≤ j ≤ d + 1.

Using the B-flag the number a_{0} imposes that

dim_{ℂ} P ∩ (p_{2m+2}⋁ ⋯ ⋁p_{d+1}) ≥ 0.

But since P lies only in p_{0}⋁ ⋯ ⋁p_{d+1} the condition imposed by a_{0} holds if and only if P contains the point p_{d+1}.

Through the same vein we argue that since the integers a_{0}, ..., a_{j−1} are consecutive the condition

dim_{ℂ} P ∩ (p_{2m+2}⋁ ⋯ ⋁p_{d+2−j}) ≥ j − 1

can hold if and only if P contains the join p_{d+1}⋁ ⋯ ⋁p_{d+2−j}. The other integers in the Schubert symbol a do not impose any further conditions on P.

In view of these arguments we find out that if the d-plane P lies in the nonempty intersection Ω_{i} · Δ_{j} then it must contain the following list of skeleton points;

p_{0}, ..., p_{d+1} and p_{d+2−j}, ..., p_{d+1}.

The first part of the list is derived from the fact that P ∈ Ω_{i} and the second part from the fact that P ∈ Q_{2m} . But there are altogether d + 1 − (i − j) skeleton points in this list and hence their join, which necessarily belongs to P, has dimension d − (i − j). Since P is a d plane we must have i − j ≥ 0 or i ≥ j.

We thus find the description for all P ∈ Ω_{i} · Δ_{j}; each such P must live in the join p_{0}⋁ ⋯ ⋁p_{d+1} and must contain a d + 1 − (i − j)-dimensional subspace of this join. This is the description of Ω_{i−j}.

This then completes the proof of the theorem. □

4. The Unstable Cases. The cases when 2d ≤ n ≤ 2d + 2 are called the unstable cases (the terminology belongs to Dibağ, see [3]). The theorem of the previous section holds verbatim in the unstable cases if we provide the correct definitions of the Δ_{j}’s. In the following subsections we describe the necessary modifications in the definitions to make the theorem hold.

4.1. The n = 2d + 2 case. In this case the Chern cycles are defined as

Ω_{i} = Ω_{0···_{(m−i)}···m_{0}} + Ω_{0···_{(m−i)}···m_{1}}, 0 ≤ i ≤ m.

Here note that m = d + 1. For these Chern cycles we define the following dual Chern cycles

Δ_{j} = Ω_{m_{0}···_{(m+j)}···n} + Ω_{m_{1}···_{(m+j)}···n}, 1 ≤ j ≤ m.

Our theorem of the previous section now holds verbatim with these definitions.

4.2. The n = 2d+1 Case. The Chern cycles for i = 0, ..., d+1 are defined as

Ω_{i} = 2Ω_{0···_{(d+1−i)}···d,d+i}.

Define the required ‘duals’ as

Δ_{j} = Ω_{d+1−j,d+1···_{(d+j)}···2d+1}, j = 0, ..., d + 1.

4.3. The n = 2d Case. This is the maximal plane case. There are two disjoint, irreducible families of d-planes in Q_{n}. Call these families V_{0} and V_{1}. The Schubert cells of G(d, Q_{n}) are evenly divided among these families. It suffices to consider V_{0} only. The V_{1} case is obtained simply by reversing the marking of d in the following definitions.

First assume that d is even. Ω_{0} is defined as

Ω_{0} = 2Ω_{0···d_{0}}.

The nontrivial Chern cycles are defined as

Ω_{i} = 2Ω_{0···_{(d−i)}···d_{1},d+i}, i = 1, ..., d.

And finally define Ω_{d+1} = 0. The ‘duals’ are then defined as

Δ_{0} = Ω_{d_{0}···2d}
Δ_{j} = Ω_{d−j,d_{1}···_{(d+j)}···2d}, j = 1, ..., d
Δ_{d+1} = 0.

When d is odd, to obtain the symbolism in V_{0} reverse the indexing of d in the definition of Chern cycles but leave the indexing of the ‘duals’ the same.

With these definitions Theorem 1 holds. Because of the significance of the maximal plane case we quote this result separately as a corollary to Theorem 1.

Corollary 2. In the n = 2d case we also have

Ω_{i} · Δ_{j} = αΩ_{i−j}

for 0 ≤ j ≤ i ≤ d where α is a nonzero integer.

Proof. We give the proof when d is even. The odd case is similar. Let Λ be a d-plane in the intersection of Ω_{i} · Δ_{j}. Assume that a fixed set of skeleton points p_{0}, …, p_{2d+1} is fixed. We interpret the Schubert conditions of Ω_{i} with respect to the A-flag and those of Δ_{j} with respect to the B-flag. Then Λ lives in p_{0}⋁ ⋯ ⋁p_{d+i+1}, must have a point p in p_{d+j+1}⋁ ⋯ ⋁p_{d+i+1}. Therefore the complex conjugate of this join, which is p_{d−i}⋁ ⋯ ⋁p_{d−j} can contribute only i − j − 1 dimension to Λ, i.e. dim (Λ ∩ (p_{0}⋁ ⋯ ⋁p_{d−j})) = d − j − 1. Since by the Schubert conditions of Ω_{i} the join p_{0}⋁ ⋯ ⋁p_{d−i−1} belongs to Λ, it follows that Λ also contains p_{d−j+1}⋁ ⋯ ⋁p_{d−1} and p_{d+1}. These conditions thoroughly describes any Λ in the intersection. To prove the corollary we translate these descriptions to Schubert conditions. For this purpose define a new set of skeleton points q_{0}, …, q_{2d+1} as follows:

• q_{t} = p_{t} where t = 0, …, d − i − 1.

• q_{d−i+t} = p_{d−j+1+t} where t = 0, …, j − 2.

• q_{d−(i−j)−1} = p_{d+1}.

• q_{d−(i−j)+t} = p_{d−i+t} where t = 0, …, (i − j) − 1.

• q_{d} = p_{d+j}. (This is to respect the V_{0}, V_{1} formalism of maximal planes in a quadric.)

• q_{d+t} = \overline{q_{d−t+1}} where t = 1, …, d + 1.

Defining an A-flag with respect to this set of skeleton points, the above description of Λ becomes equivalent to the description of Ω_{i−j} as claimed. □

5. Obstruction Classes. We first define the Chern classes as

c_{i} = Ω_{i}^{∗} = P D Ω_{i}^{t} = P D Δ_{i} ∈ H^{i}(A_{d+1}^{(n+2)}; ℤ)

where ^{∗} denotes the cell dual, P D denotes Poincaré duality and ^{t} denotes homology duality, see [3]. Since boundary operations are zero, the cycles and cocyles constitute the homology and cohomology respectively.

A necessary condition for the decomposability of a 2-form ω of constant rank 2(d + 1) on a trivial ℝ^{n+2} bundle E, with n > 2d, on some base space B is the vanishing in H^{2(d+1)}(B; ℤ) of ω_{1}^{∗}(c_{i}), 0 ≤ i ≤ d + 1, where c_{i} ∈ H^{2(d+1)}(A_{d+1}^{(n+2)}; ℤ) is the Chern class of the principal U(d+1)-bundle of Stiefel manifold of orthonormal 2(d + 1)-frames in ℝ^{n+2} with the projection onto A_{d+1}^{(n+2)} being given by (y_{1}, …, y_{2(d+1)}) ↦ y_{1} ∧ y_{d+1} ⋯ + y_{d+1} ∧ y_{2(d+1)} and where ω_{1} : B → A_{d+1}^{(n+2)} represents ω, [3].

In this section as an application of our intersection theorem we intend to describe the vanishing of ω_{1}^{∗}(c_{i}) in terms of the vanishing of ω_{1}^{∗}(P D Ω_{i}). Intersection of homology cells being Poincaré dual to cup product in cohomology we have the following relations which follow from Theorem 1:

P D Ω_{i−j} = P D (Ω_{i} · Δ_{j})
= (P D Ω_{i}) ∪ (P D Δ_{j})
= (P D Ω_{i}) ∪ c_{j}, 0 ≤ j ≤ i ≤ d + 1.

We now quote our application to obstruction of decomposability;

Theorem 3. If ω_{1}^{∗}(P D Ω_{s}) ≠ 0 for some fixed s with 0 ≤ s ≤ d + 1, then ω_{1}^{∗}(c_{i}) ≠ 0 for all i = 0, …, d + 1 − s.

Proof. This follows from the equation

ω_{1}^{∗}(P D Ω_{s}) = ω_{1}^{∗}(P D Ω_{s+i}) ∪ ω_{1}^{∗}(c_{i}).

The left hand side being nonzero, each term on the right hand side has to be nonzero. □

In particular ω_{1}^{∗}(P D Ω_{0}) is an obstruction to the vanishing of every ω_{1}^{∗}(c_{i}). We conclude with the following remark which we record as a corollary.

Corollary 4. If ω_{1}^{∗}(P D Ω_{s}) = 0 and ω_{1}^{∗}(P D Ω_{s+i}) ≠ 0 for i = 1, …, d + 1 − s, then ω_{1}^{∗}(c_{i}) = 0 for i = 1, …, d + 1 − s. In particular if ω_{1}^{∗}(P D Ω_{i}) vanishes for i = 0 only, then ω_{1}^{∗}(c_{i}) = 0 for all i = 1, …, d + 1. □

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Sinan Sertöz Bilkent University Department of Mathematics TR-06533 Ankara,Turkey

e-mail: sertoz@fen.bilkent.edu.tr
Web: http://www.fen.bilkent.edu.tr/~sertoz