DEGENERACY OF TWO AND THREE FORMS

L. M. FEHÉR, A. NÉMETHI, AND R. RIMÁNYI

Abstract. The global theory of complex 2 and 3 forms is studied in this paper. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other cohomological obstructions to the existence of ‘mildly’ degenerate forms. Based on these results and a formula for projective representations we calculate the degree of the projectivization of certain orbits of the representation Λ^{k}ℂ^{n}.

1. Introduction

1.1. 2-forms. If ω is a generic complex 2-form on a complex manifold M, then we can stratify M according to the corank of ω_{x} at x ∈ M as follows M ⊃ S_{i} := {x ∈ M| corank ω_{x} = i}— which is clearly empty when n − i is odd. Our goal is the understanding of the cohomological obstructions to ∪_{i≥r}S_{i} = ∅, for any r. That is, we want to describe the cohomological obstructions to the existence of a 2-form which everywhere drops rank by less than r. Equivalently, this set of obstructions are the classes which are “universally supported” ([FP98, Ch. 4]) on the locus \overline{S_{r}}. These classes form an ideal O_{Σ^{<r}} in the Chern classes of the manifold. They were studied originally in [Pra88] in the context of polynomials universally supported on skewsymmetric degeneracy loci, where a certain explicit description of the ideal was given using Schur P -polynomials, see also [PR96], [FP98, Ch. 4]. In Theorem 3.1 we give another explicit description of this ideal, using Schur determinants and less generators than in [Pra88]. To put the result of this Theorem in context let us suppose that ω is a non-degenerate 2-form on an even dimensional manifold M, i.e. one for which S_{0} = M. Then ω yields an isomorphism between T M and T^{∗}M, so c_{i}(T M) = c_{i}(T^{∗}M) = (−1)^{i}c_{i}(T M), that is c_{1}, c_{3}, c_{5}, … = 0. We will find that these classes generate O_{Σ^{<1}}. This is, of course, not surprising. The question is how it generalizes to greater r. In Theorem 3.1 we will show that O_{Σ^{<r}} is generated by Schur polynomials in Chern classes indexed by partitions of type odd>even>odd> . . ..

The elements of the obstruction ideal O_{Σ^{<r}} have geometric meanings. The most straightforward is the meaning of the least degree element. In the case above, it is the Poincaré dual of \overline{S_{r}} (“classes of degeneracy loci” of algebraic geometry, or “Thom polynomial” of singularity theory). Thus, taking the least degree element of the ideal above we recover results of [JLP81], [Pra90], [HT84], [FRa].

The other elements of the obstruction ideal also have geometric meanings, they are the derived Thom polynomials of [Kaz97]. We will show this phenomenon in a more spectacular case of 3-forms on 6-manifolds.

Supported by FKFP0055/2001 (1st and 3rd author), OTKA T029759 (3rd author) and NSF grant DMS-0088950 (2nd author)

Keywords: Classes of degeneracy loci, 2-forms, 3-forms, Thom polynomials, global singularity theory AMS Subject classification 14N10, 57R45.

1.2. 3-forms. Our goal is a similar analysis for 3-forms. If ω is a generic r-form on a manifold M^{n} then we can stratify M according to the orbits of the representation of GL_{n}(ℂ) on Λ^{r}(ℂ^{n}^{∗}). It is known that this representation has finitely many orbits only in the cases covered by 1.1 and in the (new) cases r = 3, n = 6, 7, 8. In this paper we will concentrate on 3-forms on 6-manifolds (and just mention some other results). There are 5 orbits of the action of GL_{6}(ℂ) on Λ^{3}(ℂ^{6}): σ_{0}, σ_{1}, σ_{5}, σ_{10} and σ_{20} (indices being the codimensions of the orbits). We will compute the obstruction ideals of σ_{0}, σ_{0} ∪ σ_{1}, σ_{0} ∪ σ_{1} ∪ σ_{5}, etc. The first is thus the collection of characteristic classes which are obstructions to the existence of a 3-form on a 6-manifold which is everywhere generic (these forms are called stable in [Hit01]). The other ideals are collections of obstructions to the existence of a 3-form which is only ‘mildly’ degenerate.

We will also give geometric characterization of a higher degree element in one of these ideals, yielding to the following theorem. If ω is a generic 3-form on a 6-manifold, and S_{5} is the Riemann surface of points x where ω_{x} ∈ σ_{5}, then the degree of the line bundle {v ∈ T_{x}^{∗}M|v∧ω_{x} = 0} → S_{5} is ∫_{M} c_{1}^{3}c_{3} + c_{1}c_{2}c_{3} + 2c_{1}^{2}c_{4} − c_{1}^{4}c_{2} − 2c_{1}c_{5} − c_{1}^{2}c_{2}^{2} where c_{i} = c_{i}(T^{∗}M).

From the singularity theoretic point of view this last phenomenon is a “derived Thom polynomial”. Indeed, we can consider a 6-manifold, for which the class q_{5} (see Theorem 4.4) vanishes but the above integral not. Then we obtain that for this manifold every 3-form must have σ_{5}-points, although the σ_{5}-points represent a 0-homologous cycle (the Thom polynomial of σ_{5} is q_{5} = 0). For more on Thom polynomials and beyond, see [SS99].

2. Review on Thom polynomials and obstruction ideals of group actions

In this section we review the notions of Thom polynomials and obstruction ideals for group actions from [FRa] (the theory of Thom polynomials is strongly motivated by [Kaz97]).

Let G act on the vector space V with finitely many orbits. Let τ be a union of orbits, usually an open one. Then the obstruction ideal of τ is defined as

O_{τ} = ker(H_{G}^{∗}(τ ⊂ V)) ◁ H_{G}^{∗}(V) = H^{∗}(BG).

This innocent definition has many advantages. First of all, it has the sought geometric meaning: elements in this ideal restrict to 0 on τ, so are supported on the complement of τ. Thus for a bundle, whose structure group is G and fiber is V, the G-characteristic classes from O_{τ} are obstructions to the existence of a section everywhere inside τ ⊂ V. The second advantage of this definition is that it is very well computable, as follows from the first two parts of the following theorem.

Theorem 2.1.

(1) If τ is one orbit, then O_{τ} = ker(H^{∗}B(G_{τ} ⊂ G)), where G_{τ} is the stabilizer (isotropy) subgroup of any element in τ;

(2) if the orbit stratification of V satisfies the Euler condition (see below), then O_{τ_{1}∪τ_{2}} = O_{τ_{1}} ∩ O_{τ_{2}};

(3) If τ is the complement of the closure of an orbit η, then H^{<codim η} ∩ O_{τ} = 0 and H^{codim η}(BG) ∩ O_{τ} is generated by the Thom polynomial (degeneracy locus) of η.

The stratification satisfies the Euler condition if the equivariant Euler classes to the orbits (in H^{∗}(BG_{η}), for the orbits η) are all non zero-divisors. This condition appeared in [AB83] as a sufficient condition for G-perfectness. The Euler condition will hold for all the representations we consider in this paper.

As can be seen from this theorem, the computation of the obstruction ideals only needs standard techniques of algebraic topology, so can be done in every concrete case. The third part of the theorem says that this computability also applies to the Thom polynomials of the orbits. (This was the way the classical Thom polynomials were computed in [Rim01].)

Let us also remark that a consequence of the Theorem above is that in order to carry out the calculations we need to determine the stabilizer subgroups only up to homotopy equivalence (i.e. we can work with U(n) instead of GL_{n}(ℂ), etc.).

3. Degeneracy of 2-forms

In this section we apply the theory of Section 2 to the representation Λ^{2}(ℂ^{n}) of GL(n, ℂ). In this paper we work over the complex field ℂ so we also use the shorthand notation GL(n) for GL(n, ℂ) etc. As it is well known, here the orbits are characterized by corank r, i.e. every

2-form is equivalent to one of the following matrices \frac{n−r}{2}H ⊕ 0_{r×r}, where H = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}). Let

the orbit with corank r be called Σ^{r}. Its codimension is \binom{r}{2} if r ≥ 2 and is 0 if r = 0 (n even) or r = 1 (n odd).

Theorem 3.1. Let Δ_{λ} denote the Schur polynomial associated with the partition λ as in [FP98, (1.5)], evaluated at the universal Chern classes of BGL(n), and let n − r be even. Then

(1) O_{Σ^{<r}} = ⟨Δ_{i_{r−1},i_{r−2},...,i_{2},i_{1}}|i_{r−1} > i_{r−2} > … > i_{1}, i_{odd} is odd, i_{even} is even⟩;

(2) Tp(Σ^{r}) = Δ_{r−1,r−2,...,2,1}, if r ≥ 2 (otherwise Tp = 1).

Proof. Let n−s be even. The stabilizer subgroup of a representative from Σ^{s}, e.g. the one given above, is clearly G_{s} = Sp_{\frac{n−s}{2}} × GL_{s} (modulo homotopy equivalence), with H^{∗}BG_{s} = ℤ[p_{i}, a_{j}] where i = 1, …, \frac{n−s}{2} index the characteristic (Pontryagin) classes of Sp(\frac{n−s}{2}) and j = 1, …, s index the Chern classes of GL(s). Let us use the convention that c_{<0}, a_{<0}, p_{<0}, c_{>n}, a_{>s}, p_{> \frac{n−s}{2}} vanish and c_{0} = a_{0} = p_{0} = 1. From the inclusion G_{s} ⊂ G we can also read off the induced ring homomorphism H^{∗}(BG) = ℤ[c_{1}, …, c_{n}] → H^{∗}(BG_{s}) = ℤ[p_{i}, a_{j}]:

c_{i} ↦ ∑_{j=0}^{\frac{n−s}{2}} p_{j}a_{i−2j}.

We claim that the kernel of this homomorphism, i.e. O_{Σ^{s}}, is the ideal generated by the Schur polynomials Δ_{i_{s+1},i_{s},...,i_{2},i_{1}} where i_{s+1} > i_{s} > … > i_{1} and i_{odd} is odd, i_{even} is even. This can be seen by eliminating the variables p_{j} and a_{i} from the equations induced by

∑_{i=1}^{n} c_{i}t^{i} = (∑_{i=1}^{s} a_{i}t^{i})(∑_{j=1}^{\frac{n−s}{2}} p_{j}t^{2j}).

If we write down this equation with t and −t as indeterminate, and consider the sum and difference of these equations, we obtain

(∑_{2i≤n} c_{2i}t^{2i})( ∑_{2i+1≤s} a_{2i+1}t^{2i+1}) − ( ∑_{2i+1≤n} c_{2i+1}t^{2i+1})(∑_{2i≤s} a_{2i}t^{2i}) = 0.

Expanding and rearranging the terms we obtain the system of linear equations

((−1)^{j}c_{2i−j})_{i=1,..., \frac{n+s}{2} ,j=1,...,s+1}(a_{j−1})_{j=1,...,s+1} = 0,

which has solutions exactly when the maximal minors of the coefficient matrix ((−1)^{j}c_{2i−j}) vanish. These minors are Δ_{i_{s+1},i_{s},...,i_{2},i_{1}} with i_{s+1} > i_{s} > … > i_{1} and i_{odd} is odd, i_{even} is even, so we obtain that

O_{Σ^{s}} = ⟨Δ_{i_{s+1},i_{s},...,i_{2},i_{1}} | i_{s+1} > i_{s} > … > i_{1}; i_{odd} is odd, i_{even} is even⟩.

Because of this description one can use expansions of determinants to see that if s_{1} > s_{2} then O_{Σ^{s_{1}}} ⊂ O_{Σ^{s_{2}}}, so according to Theorem 2.1 we have

O_{Σ^{<r}} = ⋂_{s<r} O_{Σ^{s}} = O_{Σ^{r−2}} = ⟨Δ_{i_{r−1},i_{r−2},...,i_{2},i_{1}} | i_{r−1} > i_{r−2} > … > i_{1}; i_{odd} is odd, i_{even} is even⟩.

The proof of the first statement is complete. The other statement —after checking sign— follows from the general theory that the Tp is the least degree element in the corresponding obstruction ideal.

Corollary 3.2. Let ω be a generic 2-form on M, with c_{i} = c_{i}(T^{∗}M). Then the cohomology class represented by the set \overline{S_{r}} of points x where ω_{x} drops rank by at least r is Δ_{r−1,r−2,...,2,1}. If any element in the ideal above is not 0 then the set \overline{S_{r}} can not be empty.

Remark 3.3. The second part of Theorem 3.1 has already been known, see [JLP81], [HT84], and [FRa]. Another description of the ideal using Schur P -polynomials and more generators was given in [Pra88], see also [PR96], [FP98, Ch.4].

Remark 3.4. There is geometric meaning not only of the least degree element in the ‘avoiding’ ideal above, but also of the higher degree ones. These geometric interpretations are quite artificial in this case, so we do not give the details, but will show how to deal with such questions in a more illuminating case, see Theorem 4.5.

4. Degeneracy of 3-forms on 6-manifolds

Now let us turn to the representation Λ^{3}(ℂ^{6}) of GL(6). The description of the orbits were known by Segre, for a modern account see [Don77]. For us the geometric description is not needed, only representatives from each orbit, which can be chosen using Donagi’s description.

Theorem 4.1. Let e_{1}, …, e_{6} form a basis of ℂ^{6}. The representation Λ^{3}(ℂ^{6}) of GL_{6}(ℂ) has 5 orbits σ_{0}, σ_{1}, σ_{5}, σ_{10}, σ_{20} (indices are the codimensions), with representatives

ω_{0} = e_{123} + e_{456}  ω_{1} = e_{126} + e_{135} + e_{234}  ω_{5} = e_{1} ∧ (e_{23} + e_{45})  ω_{10} = e_{123}  ω_{20} = 0,

where e_{ij...} means e_{i} ∧ e_{j} ∧ . . ..

To apply our general theory we need to know (at least up to embedded homotopy equivalence) the stabilizer subgroups of these representatives. In each cases it is an easy computation (with ω_{0} being probably the hardest, but that one is given in [Hit00]).

Theorem 4.2. The following groups are (modulo embedded homotopy equivalence) the stabilizer subgroups of the above representatives (1) [Hit00] G_{0} = (SL(3) × SL(3)) ⋊ ℤ_{2}. The two SL(3)’s act on e_{1}, e_{2}, e_{3} and e_{4}, e_{5}, e_{6}

respectively, and ℤ_{2} interchanges these two ℂ^{3}’s.

(2) G_{1} = U(1)^{3}⋊S_{3}. For α, β, γ ∈ U(1)^{3} the action is via the diagonal matrices (α, β, γ, β̄γ̄, γ̄ᾱ, ᾱβ̄). The symmetric group S_{3} permutes e_{1}, e_{2}, e_{3} and e_{4}, e_{5}, e_{6} simultaneously.

(3) G_{5} = U(1)^{2} × Sp(2). For (α, β) ∈ U(1)^{2} the action is (α, ᾱ, 1, ᾱ, 1, β). Sp_{2} acts on the e_{2}, e_{3}, e_{4}, e_{5} the standard way.

(4) G_{10} = SL(3)×GL(3). The group SL(3) acts on e_{1}, e_{2}, e_{3} and GL_{3} acts on the remaining coordinates.

(5) G_{20} = GL(6).

Remark 4.3. The theory of Thom polynomials and obstruction ideals itself serves as a check whether we found all the symmetries. Indeed, the orbit stratification of Λ^{3}ℂ^{6} induces a filtration of this vector space, which yields the Kazarian spectral sequence (the GL(6)-equivariant relative cohomology spectral sequence) converging to H^{∗}(BGL_{6}). The stratification is GL(6)-perfect, therefore this spectral sequence degenerates at E_{1}^{∗,∗}, so we must have

dim H^{i}(BGL_{6}) = ∑_{j∈{0,1,5,10,20}} dim H^{i−j}(BG_{j}),

cf. [FRa, Sect. 10]. In fact this observation is used to find the Betti numbers of one stratum, when those of the others are known in an infinite dimensional representation in [AB83].

Now we have all the input to compute the obstruction ideals. To save space we only write down the ones we need for the corollary after the theorem. (That is, we will omit everything of degree > 6).

Theorem 4.4. The following are obstruction ideals of ω_{0} and ω_{0} ∪ ω_{1} modulo terms of degree > 6 and using rational coefficients,

(1) O_{ω_{0}} = ⟨c_{1}⟩,

(2) O_{ω_{0}∪ω_{1}} = ⟨q_{5}, q_{6}⟩, where q_{5} = c_{1}^{5} + c_{1}c_{2}^{2} + 2c_{1}^{2}c_{3} − 4c_{1}c_{4}, q_{6} = c_{1}^{3}c_{3} + c_{1}c_{2}c_{3} + 2c_{1}^{2}c_{4} − c_{1}^{4}c_{2} − 2c_{1}c_{5} − c_{1}^{2}c_{2}^{2}.

Proof. The ideal O_{ω_{0}} is the kernel of the homomorphism H^{∗}BGL_{6} → H^{∗}BG_{0}. This homomorphism, according to the description above, is ℚ[c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}] → ℚ[a_{2}, a_{3}, b_{2}, b_{3}],

c_{1} ↦ 0 c_{2} ↦ a_{2} + b_{2} c_{3} ↦ a_{3} + b_{3} c_{4} ↦ a_{2}b_{2} c_{5} ↦ a_{2}b_{3} + a_{3}b_{2} c_{6} ↦ a_{3}b_{3}.

The kernel of this homomorphism is the ideal ⟨c_{1}, c_{3}^{2}c_{4} −c_{2}c_{3}c_{5} +c_{5}^{2} +c_{2}^{2}c_{6} −4c_{4}c_{6}⟩, which proves the first statement. (Here and in other concrete algebraic calculations of this paper we used the computer algebra package Macaulay2 [GS]). To prove the second statement we need to intersect this ideal with the kernel of the map ℚ[c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}] → ℚ[a, b, c]^{S_{3}}, given by mapping c_{i} to the i’th elementary symmetric polynomial of a, b, c, −b − c, −c − a, −a − b. Computation shows that this intersection is generated by degree 5, 6, 7, 10 polynomials, the first two being q_{5} and q_{6} from the Theorem.

We used rational coefficients because of the ℤ_{2} factor in the first stabilizer group. The disadvantage is that thus we only obtained that the Thom polynomials (degeneracy loci) of ω_{1} and ω_{5} are rational multiples of c_{1} and q_{5}. In fact, these rational multiples can be determined using the so called “principal equation” on Theorem 3.5 of [FRa]: Tp(ω_{1}) = 2c_{1}, Tp(ω_{5}) = q_{5}.

As always, the least degree elements in the obstruction ideals are the Thom polynomials. Higher degree elements are the derived Thom polynomials of [Kaz97]. Using this we will show the geometry of the generator q_{6}.

Theorem 4.5. Let ω be a generic 3-form on a 6-manifold M. Then the set S ⊂ M of points where ω is equivalent to the normal form ω_{5} (see above) is a smooth Riemann surface. Then {v ∈ T_{x}^{∗}M|x ∈ S, v ∧ ω_{x} = 0} → S is a line bundle, whose degree is q_{6}(c(T^{∗}M)).

Proof. The bundle Λ^{3}(T^{∗}M) has structure group GL_{6}(ℂ) over M. However when we restrict this bundle to S the structure group reduces to G_{5} = U(1)^{2} × Sp_{2}, with different characteristic classes. For instance this restricted bundle has two degree 2 characteristic classes a and b corresponding to the two copies of U(1). Since S is smooth in our case (the next orbit comes in codimension 10) we can consider the (Gysin) push-forward of any linear combination of a and b. These will clearly be in the degree 6 part of O_{ω_{0}∪ω_{1}}, i.e. ℚ · c_{1}q_{5} + ℚ · q_{6}. (In fact, a look at the Kazarian spectral sequence shows that they will span it.)

So we need to compute i_{!}a and i_{!}b for i : BG_{5} ⊂ BGL_{6}. We will use an extension of the method of “restriction equations” of [FRa] by restricting the equation i_{!}a = α·c_{1}q_{5}+β·q_{6} to BG_{5}. If we use the notation H^{∗}(Sp_{2}) = ℤ[k, l]^{D_{4}} (D_{4} is generated by k ↔ −k, l ↔ −l, k ↔ l) then the Chern classes restrict to the elementary symmetric polynomials of a, k − a, −k, l − a, −l, b. This gives the restriction of the left hand side. To compute the restriction of the right hand side we determine the normal direction of the orbit of ω_{5} at ω_{5}: e_{246}, e_{256}, e_{346}, e_{356}, e_{456}. So the left hand side restricts to i^{∗}i_{!}a = a · i^{∗}i_{!}1 = a · e where e is the equivariant Euler class of the normal space to ω_{5}, i.e. e = (l + k − 2a + b)(−l + k − a + b)(−k + l − a + b)(−k − l + b)(−a + b). If we write this out, it is a system of linear equations in α and β with the only solution α = 0, β = 1. So we obtain that i_{!}a = q_{6}. (Similarly we would obtain that i_{!}b = c_{1}q_{5} + q_{6}.) Since a is the first Chern class of the line bundle over S corresponding to the e_{1} direction, and this e_{1} direction can be characterized as {v|v ∧ ω_{5} = 0}, the theorem follows.

Remark 4.6. Theorem 4.5 shows geometrically that q_{6} is indeed in the ideal O_{ω_{0}∪ω_{1}}, just like q_{5}, for which q_{5}(c(T^{∗}M)) = [S] ∈ H^{5}(M). The extra information in the description of the ideal is that these two generators are enough.

5. Some sporadic results

In this section we present some sporadic results on obstructions to the existence of certain forms on 7 and 8 manifolds. These results follow easily from standard algebraic topological techniques—the reason of giving them is that they fit naturally to the earlier results of this paper on one hand, and they have some relevance in the theory of manifolds with special holonomy [Joy00] on the other.

5.1. Stable 3-forms on 7 manifolds. The representation Λ^{3}(ℂ^{7}) of GL(7) has finitely many orbits, in particular there is an open orbit σ_{0}. As a consequence, it makes sense to talk about 3-forms on a closed 7 manifold which are generic everywhere (stable 3-forms [Hit01]). The elements of the obstruction ideal of σ_{0} (evaluated at the Chern classes of the cotangent bundle) are thus obstructions to the existence of stable 3-forms on M.

Theorem 5.1. Using rational coefficients the obstruction ideal of the open orbit of the representation Λ^{3}(ℂ^{7}) of GL(7) is O_{σ_{0}} = ⟨c_{1}, c_{3}, c_{5}, c_{7}, c_{2}^{2} − 4c_{4}⟩.

Proof. It is well known that the stabilizer subgroup of a generic 3-form on ℂ^{7} is the exceptional Lie group G_{2} × ℤ_{3} [Her83], where G_{2} acts by the representation with highest weight 2·short root+long root, see [FH91, 22.3]. So the sought obstruction ideal is the kernel of the map from ℚ[c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}, c_{7}] to ℚ[a, b] mapping c_{i} to the i^{th} elementary symmetric polynomial of the roots of this representation, i.e. of 0, a, −a, b + a, −b − a, b + 2a, −b − 2a. The kernel is the ideal above.

Corollary 5.2. The elements of this ideal evaluated at the Chern classes of the cotangent bundle of M^{7} are obstructions to the existence of a (complex) G_{2}-structure on M^{7}. (On G_{2}-structure on M^{7} we mean a reduction of the structure group of T M to G_{2}.)

5.2. Stable 3-forms on 8 manifolds. There are finitely many orbits of the representation Λ^{3}(ℂ^{8}) of GL(8), so there is an open orbit here, too. We can work out the obstruction ideal in this case. The computation is very similar to that in subsection 5.1 with the only difference that we need to use the adjoint representation corresponding to the root system A_{2}. Since the stabilizer subgroup here P SL(2) is not simply connected we only get the result with rational coefficients.

Theorem 5.3. Using rational coefficients the obstruction ideal of the open orbit of the representation Λ^{3}(ℂ^{8}) of GL_{8}(ℂ) is O_{σ_{0}} = ⟨c_{1}, c_{3}, c_{5}, c_{7}, c_{8}, c_{2}^{2} − 4c_{4}⟩.

5.3. Spin(7) structure on 8-manifolds. The existence of a (complex) Spin(7) structure on an 8 manifold M is equivalent to the existence of a certain degenerated 4-form on M with stabilizer subgroup Spin(7), see [Joy, 1.3]. Here the representation of Spin(7) is the one whose highest weight is the long root of the root system B_{3}. The weights of this representation are γ, β − γ, α − β + γ, −α + γ and their opposites. So the obstruction ideal is the kernel of the map from ℤ[c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}, c_{7}, c_{8}] to ℤ[α, β, γ] mapping c_{i} to the ith elementary symmetric polynomial of the above 8 weights. So we obtain

Theorem 5.4. The obstruction ideal of the orbit of Ω_{0} of [Joy, 1.3] in the representation Λ^{4}(ℂ^{8}) of GL_{8}(ℂ) is O_{Ω_{0}} = ⟨c_{1}, c_{3}, c_{5}, c_{7}, c_{2}^{4} − 8c_{2}^{2}c_{4} + 16c_{4}^{2} − 64c_{8}⟩.

Corollary 5.5. The elements of this ideal evaluated at the Chern classes of the cotangent bundle of M^{8} are obstructions to the existence of a (complex) Spin(7)-structure on M^{8}.

Remark 5.6. The Thom polynomial as well as the obstruction ideal results can be translated from the complex to the real case using classical techniques of [BH61] as it is e.g. done in a different case in [FRb]. However to get reasonable results one needs the complex results over ℤ not over ℚ. Also, the obtained real results often follow from the existence of well known extra (almost complex, spin) structures.

6. Projective Thom polynomials and degree calculations

If a group G acts linearly on a vector space V then this action ρ induces an action ℙρ of G on the projective space ℙV. If the image of ρ contains the scalars then there is a bijection between the orbits of ℙρ and the non zero orbits of ρ. Strangely enough the projective Thom polynomials formally contain more informations than the affine ones: Suppose that η is a d complex codimensional invariant subset of ρ and ℙη is the corresponding invariant subset of ℙρ. Then Tp(ℙη) is an element in H_{G}^{2d}(ℙV) ≅ H^{2d}(BG)[ξ]/ ∏ ξ − β_{i} where β_{i} are the weights of the representation ρ. So Tp(ℙη) = ∑ p_{i}ξ^{i} where p_{i} ∈ H^{2(d−i)}(BG). It is easy to see that p_{0} is the affine Thom polynomial Tp(η) and p_{d} is the degree of the closure of ℙη in ℙV:

(1)  p_{0} = Tp(η)  p_{d} = deg(ℙη)

In fact the main application of the projective Thom polynomial is that we can calculate the degree of certain varieties.

The main result of this section is that a simple substitution into the affine Thom polynomial Tp(η) provides the projective Thom polynomial. To state the result we need to give names to the generators of H^{∗}(BG). Let m : U(1)^{n} → G be a (coordinatized) maximal torus of G and let α_{i} are the corresponding roots. So by the Borel theorem (or splitting principle) Tp(η) is a polynomial in the roots α_{i} and Tp(ℙη) is a polynomial in the roots α_{i} and ξ. We assume that the image of ρ contains the scalars i.e. there is a homomorphism φ : GL(1) → G and a non zero integer q such that ρ ∘ φ(λ) = λ^{q}v for all v ∈ V, λ ∈ GL(1). We assume that Im m ⊃ Im φ|_{U(1)} so we have a homomorphism φ̃ : U(1) → U(1)^{n} such that φ|_{U(1)} = m ∘ φ̃. The homomorphism φ̃ is necessarily of the form φ̃(t) = (t^{w_{1}}, …, t^{w_{n}}) where t ∈ U(1) and w_{i} are integers. Notice that the choice of φ is not unique.

Theorem 6.1. Let ρ : G → GL(V) be a representation of the Lie group G such that the image of ρ contains the scalars. Let α_{i}, q, w_{i} be as above and let η be an invariant subset of ρ. Then

Tp(ℙη)(α_{1}, …, α_{n}, ξ) = Tp(η)(α_{1} + \frac{w_{1}}{q} ξ, …, α_{n} + \frac{w_{n}}{q} ξ).

Proof. The trick is that we relate the projective Thom polynomial to an affine Thom polynomial for a different group. Let G̃ = GL(1) × G and ρ̃ = Hom(ρ^{1}, ρ) acting on Hom(ℂ, V) (where ρ^{1} denotes the standard representation of GL(1) on ℂ). Since Hom(ℂ, V) is naturally isomorphic to V we can think G̃ acting on V. Moreover the invariant subsets of ρ̃ are the same as of ρ. We have a map Q : H^{∗}(BG̃) → H_{G}^{∗}(ℙV) induced by the classifying map of the ρ̃-bundle Hom(τ, B_{G}V) → B_{G}ℙV. Here B_{G} denotes the Borel construction i.e. B_{G}ℙV = EG ×_{G} ℙV, τ is the tautological line bundle and with some abuse of notation we denote by B_{G}V the pull back of the universal ρ-bundle. It is easy to see that H^{∗}(BG̃) ≅ H^{∗}(BG)[ξ] and Q is simply the factorization with the relation ∏ ξ − β_{i}. Next we need the following

Proposition 6.2. For any invariant subset η of ρ

Q(TpG̃(η)) = Tp^{G}(ℙη).

Proof. The bundle Hom(τ, B_{G}V) has a canonical section σ, coming from the inclusion of the fiber of the tautological bundle into the vector space V. By definition η(σ) = B_{G}ℙη. So the claim reduces to the definition of Tp^{G}(ℙη).

So we can concentrate on calculating TpG̃(η). To do that we choose the maximal torus of G̃ of the form m̃(t_{0}, t_{1}, …, t_{n}) = (t_{0}, m(t_{1}, …, t_{n})). The key observation is that restricted to these maximal tori the representation ρ̃ “almost” factors through ρ. We define a homomorphism κ : U(1)^{n+1} → U(1)^{n}:

κ(t_{0}, t_{1}, …, t_{n}) := (t_{0}^{w_{1}}t_{1}^{q}, …, t_{0}^{w_{n}}t_{n}^{q}) = φ̃(t_{0})(t_{1}^{q}, …, t_{n}^{q}).

Then ρ ∘ κ(t_{0}, t_{1}, …, t_{n}) = ρ(φ̃(t_{0})(t_{1}^{q}, …, t_{n}^{q})) = ρ(φ̃(t_{0})ρ(t_{1}^{q}, …, t_{n}^{q})) = (t_{0} · ρ(t_{1}, …, t_{n}))^{q} = (ρ̃ ∘ m̃(t_{0}, t_{1}, …, t_{n}))^{q}.

Now we use the following general fact:

Proposition 6.3. Let h : K → G be a homomorphism of Lie groups and σ := ρ ∘ h. Then (1) if η ⊂ V is ρ-invariant, then η is σ-invariant, too.

(2) Tp^{σ}(η) = (Bh)^{∗} Tp^{ρ}(η) where Bh : BK → BG is induced by h.

Proposition 6.3 is an obvious consequence of the fact that the universal σ-bundle is the pull back of the universal ρ-bundle via Bh (you may consider this as the definition of Bh).

Applying Proposition 6.3 twice, first to h = κ, (Bh)^{∗}(α_{i}) = qα_{i} + w_{i}ξ then to h(x) = x^{q}, (Bh)^{∗}(α_{i}) = qα_{i} finishes the proof of Theorem 6.1.

Corollary 6.4. Using the notation of Theorem 6.1

deg(ℙη) = Tp(η)(w_{1}/q, …, w_{n}/q).

6.1. Applications. We proved that the Thom polynomial of an orbit determines the projective Thom polynomial. The coefficient of the ξ^{d} term of the projective Thom polynomial is the degree of the (projectivized) orbit. So knowing the Thom polynomial we can calculate the degree of the projectivized orbits by substituting w_{i}/q into the Chern roots. Let us illustrate how it works with some examples.

Example 6.5. Consider the action of GL_{n}(ℂ)×GL_{p}(ℂ) on Hom(ℂ^{n}, ℂ^{p}), given by (A, B)·X := BXA^{−1}. Orbits correspond to coranks. The projectivized orbit of corank r matrices is exactly the so-called corank-r determinantal variety. The Thom polynomial of Σ^{r} is the so-called Giambelli-Thom-Porteous formula, and one can choose q = 1, w_{i} = 0, 0, …, 0, 1, 1, …, 1 (or q = 1, w_{i} = (−1, −1, …, −1, 0, 0, …, 0)). Thus our theorem recovers the formula for the degree of determinantal varieties given in [Ful98, 14.4.14].

Example 6.6. All the Thom polynomial computations of this paper can be translated to degree calculations. In particular we can recover the calculations of [HT84] (about an inaccuracy of that paper see [FP98, p.78]) for the degree of the degeneracy loci S̄_{r}. For simplicity we give two special cases.

Proposition 6.7. If n is even then the dual of the Grassmannian Gr_{2}(ℂ^{n}) in P (Λ^{2}ℂ^{n}) has codimension 1 and degree n/2 [Las81], [Hol79]. If n is odd then the dual of the Grassmannian Gr_{2}(ℂ^{n}) in P (Λ^{2}ℂ^{n}) has codimension 3 and degree n(n + 1)(n − 1)/24 (for small values of n see [Hol79]).

Proof. The cone over the Grassmannian (via the Plücker embedding) is the smallest stratum, i.e. Σ^{n−2}, so the cone over its dual is the largest stratum, i.e. Σ^{2} if n even and Σ^{3} if n is odd. The Thom polynomials of these are Δ_{1} = c_{1} and Δ_{2,1} = c_{1}c_{2} − c_{3} respectively. Their degree thus gives the codimension of the dual of the Grassmannians: 1 and 3 in the two cases. According to Corollary 6.4 we get the degrees if we substitute 1/2 into the Chern roots. Hence if n is even then the degree is \frac{1}{2} + … + \frac{1}{2} (n terms) = n/2, while if n is odd then degree is

det ( 1 n\frac{1}{2}^{\binom{n}{2} \frac{1}{4} \binom{n}{3} \frac{1}{8}} ) = n(n + 1)(n − 1)/24.

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Department of Analysis, Eotvos University, Budapest

E-mail address: lfeher@math-inst.hu

Department of Mathematics, The Ohio State University

E-mail address: nemethi@math.ohio-state.edu

Department of Mathematics, The Ohio State University

E-mail address: rimanyi@math.ohio-state.edu