The even Clifford structure of the fourth Severi variety

The Hermitian symmetric space $M=\mathrm{EIII}$ appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle $E$ over it together with an algebra bundle morphism $\varphi:\mathrm{Cl}^0(E) \rightarrow \mathrm{End}(TM)$ mapping $\Lambda^2 E$ into skew-symmetric endomorphisms, and the existence of a metric connection on $E$ compatible with $\varphi$. We give an explicit description of such a vector bundle $E$ as a sub-bundle of $\mathrm{End}(TM)$. From this we construct a canonical differential 8-form on $\mathrm{EIII}$, associated with its holonomy $\mathrm{Spin}(10) \cdot \mathrm{U}(1) \subset \mathrm{U}(16)$, that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at $\mathrm{EIII}$ as the smooth projective variety $V_{(4)} \subset \mathbb{C}P^{26}$ known as the fourth Severi variety.

transformations preserving det A = 1 6 (trace A) 3 − 1 2 (trace A)(trace A 2 ) + 1 3 trace A 3 turns out to be the exceptional simple complex Lie group E 6 (C). The action of E 6 (C) on the associated projective space CP 26 = P (H 3 (C ⊗ O)) has three orbits, defined by the possible values of the rank of matrices. The closed orbit, consisting of rank one matrices and defined by the quadratic equation Next, we look at EIII in another aspect, namely as a smooth complex projective algebraic variety. In this respect EIII has been called the fourth Severi variety. This term refers more generally to the possibility of defining projective planes over four complex composition algebras, namely over C ⊗ R, C ⊗ C, C ⊗ H, C ⊗ O, and embedded in complex projective spaces of suitable dimension. One can in fact construct in a unified way (cf. [LM01]) projective planes over the four listed composition algebras, and get in this way the four Severi varieties V (1) , V (2) , V (3) , V (4) as smooth complex projective varieties respectively in CP 5 , CP 8 , CP 14 , CP 26 . Both the ambient spaces and the Severi varieties can be seen as projectified objects, the former of the Jordan algebra of Hermitian matrices, and the latter of their sets of rank one matrices. Further informations on the Severi varieties will be given in Section 3.

Preliminaries
A natural approach to Spin(10)-structures is via an extension of the following notion, used in real 16dimensional Riemannian geometry (see [Fri01] for Spin(9)-manifolds, and [PP12] for some applications). In the terminology of the Introduction, E 9 is a non-essential even Clifford structure, defined through the local Clifford systems (I 1 , . . . I 9 ). From these data one gets on M the local almost complex structures J αβ = I α • I β , and the 9 × 9 skew-symmetric matrix of their Kähler 2-forms (2.3) ψ C = (ψ αβ ).
The following is proved in [PP12]: be the skew-symmetric matrix of the Kähler 2-forms associated with the family of complex structures J αβ . If τ 2 and τ 4 denote the second and fourth coefficient of the characteristic polynomial, then: where Φ Spin(9) ∈ Λ 8 (R 16 ) is the canonical form associated with the standard Spin(9)-structure in R 16 .

The fourth Severi variety V (4) ⊂ CP 26
The following characterization of the four Severi varieties was proved by F. L. Zak in the early 1980s in the context of chordal varieties ( [Zak85,LVdV84]). Let V n ⊂ CP m be a smooth complex projective variety not contained in a hyperplane, and assume that its dimension n satisfies n = 2 3 (m − 2). Then the chordal variety Chord V , locus of of all the secant and tangent lines, coincides with CP m , unless n = 2, 4, 8, 16 and V is one of the following projective varieties: , the Plücker embedding of this Grassmannian in CP 14 , iv) V (4) ∼ = EIII, the projective plane over complex octonions as a smooth subvariety of CP 26 . Moreover, the lower codimension hypothesis n > 2 3 (m − 2) insures that Chord V = CP m . For the four mentioned exceptions, namely the Severi varieties V (i) (i = 1, 2, 3, 4), the chordal variety Chord V (i) coincides with the cubic hypersurface det A = 0, i.e. with the variety of matrices of rank ≤ 2 in the construction via the Jordan algebra H 3 in the respective complex composition algebra.
The name for these four V (i) was given by Zak in recognition of a 1901 F. Severi's work [Sev01], investigating projective surfaces with the mentioned chordal property, and characterizing in this way the Veronese surface V (1) of CP 5 . It is notable that from Zak classification it follows that all the four Severi varieties can be looked at "Veronese surfaces", i.e. at projective planes embedded in complex projective spaces via an appropriately written "Veronese map" wherex m denotes the conjugation in the second factor algebra (cf. [Zak85, Theorems 6 and 7]). It is relevant for us that the four Severi varieties appear in the following table of "projective planes" (K ⊗ K ′ )P 2 : Gr or 4 (R 12 ) E 7 /Spin(12) · Sp(1) O OP 2 ∼ = F 4 /Spin(9) E 6 /Spin(10) · U(1) ∼ = V 78 16 E 7 /Spin(12) · Sp(1) E 8 /Spin(16) + This can be seen as an application to compact symmetric spaces of the Freudenthal magic square of Lie algebras, see e.g. [Bae02,page 193]. In particular, in the C-row and the C-column of the above table we see the four Severi varieties where following the classical notations V d n denotes a complex projective algebraic variety of dimension n and degree d in a CP m .
We can also recognize how the cohomology of the first, second and third Severi variety is generated by the cohomology classes of canonical differential forms. There is of course the Kähler 2-form as the only generator on V (1) ∼ = CP 2 , and there are the two Kähler 2-forms of the factors for V (2) ∼ = CP 2 × CP 2 . On V (3) ∼ = Gr 2 (C 6 ) the cohomology generators are the complex Kähler 2-form ω and the quaternionic 4-form Ω, since Gr 2 (C 6 ) turns out to have both a complex Kähler and a quaternion-K"ahler structure, with no compatibility between them. Thus one expects something similar for the fourth Severi variety V (4) ∼ = EIII, where its complex Kähler structure may be non-compatible with its "octonionic" one.
Both the cohomology algebra and the Chow ring of V (4) ∼ = EIII have been computed (see [TW74,IM05,DZ10]). The integral cohomology algebra has no torsion and: where a 1 ∈ H 2 , a 4 ∈ H 8 and r β denote relations in H 2β . A CW-decomposition of EIII into Schubert cycles is described in [IM05,DZ10]. This allows to get the Chow ring of EIII, whose structure is isomorphic to that of mentioned cohomology. This has been done in [IM05] by obtaining three generators and several relations, and in [DZ10] has been observed that two generators suffice. The following picture, taken from [IM05], describes the Schubert cycles of EIII, labelled by their degrees, and their incidence relations. The complex dimension of the cycles goes from zero on the left to 16 on the right, where the full fourth Severi variety V 78 16 appears. The four black nodes appearing in the the diagram emphasize, besides the whole variety EIII ∼ = (C ⊗ O)P 2 ∼ = V 78 16 , its totally geodesic "projective line" (C⊗O)P 1 ∼ = Gr 2 (R 10 ), isometric to a non singular quadric Q 8 ⊂ CP 9 . It is well known from projective geometry (see for example [Seg72, page 64]), that even dimensional non singular quadrics admit two families of maximal linear subspaces. In the case of Q 8 these are two 10-dimensional families of 4-dimensional linear subspaces. Elements CP 4 , (CP 4 ) ′ of these two families appear in the diagram, where it appears that the (CP 4 ) ′ (but not the CP 4 ) are extendable to 5-dimensional linear subspaces in V 78 16 , but as mentioned non-extendable in Q 8 .
Remark 3.1. The two families (both of complex dimension 10) of complex projective spaces CP 4 ⊂ Q 8 can be viewed also as the families of sub-Grassmannians Gr 1 (C 5 ) ⊂ Gr 2 (R 10 ) with respect to a choice of complex structures on the two families parametrized by the Hermitian symmetric space SO(10)/U(5), with respect to the two possible orientations. This observation will be used in the proof of Theorem 1.2.

A Clifford system and a Lie subalgebra h ⊂ so(32)
The construction outlined in Section 2 for the canonical 8-form Φ Spin(9) can be seen in parallel with those of other canonical differential forms.
In particular, the datum of a rank 5 vector bundle E 5 ⊂ End(T M ) over a Riemannian manifold M 8 , locally generated by involutions I 1 , . . . , I 5 satisfying properties (2.1), (2.2), is equivalent to the datum of an almost quaternion-Hermitian structure of M 8 . One sees in particular that the quaternionic 4-form in real dimension 8 can be constructed from E 5 , cf. [PP12, page 329]. On the other hand, the vector bundles E 5 ⊂ End(T M ) and E 9 ⊂ End(T M ), when M is respectively a Riemannian M 8 or M 16 , are examples of even Clifford structure in the sense of [MS11], and they are both non-essential, according to the definition given in the Introduction.
Thus, in the mentioned examples, such an even Clifford structure is equivalent to the datum on M 8 or M 16 of a Sp(2) · Sp(1) or a Spin(9)-structure.
It is therefore natural to inquire about the possibility of a similar approach for a Spin(10)-structure on C 16 , and again more generally on manifolds. The following Proposition shows that the same approach cannot be pursued without modification for the group Spin(10).
Proposition 4.1. The complex space C 16 does not admit any family of ten endomorphisms I 0 , . . . , I 9 , satisfying the properties (2.1) and (2.2) with respect to the standard Hermitian scalar product g.
Proof. Assume that I 0 , . . . , I 9 are involutions on C 16 satisfying (2.1) and (2.2). Note that any such I α lies necessarily in U(16). To show the non-existence on C 16 of such a datum, we will see that the set of all compositions of two, three and six such involutions would give rise to linearly independent complex structures on C 16 . In fact, counting their number, we would obtain in this way 45+120+210=375 linearly independent complex structures. But the dimension of the space Λ 1,1 of orthogonal complex structures on C 16 is only 256.
Going into some details, it is an easy consequence of relations (2.1) and (2.2) that any composition of 2, 3 or 6 different involutions among I 0 , . . . , I 9 is a complex structure, whereas the composition of 4 or 5 of them is an involution. With this in mind, we can show that the complex structures listed in (4.1) are mutually orthogonal. To see this, and as a first observation, we get immediately that tr(J * αβ J γδ ) = tr(I β I α I γ I δ ) = 0 if any of γ < δ equals any of α < β. But also tr(J * αβ J γδ ) = tr(I β I α I γ I δ ) = 0 if α = γ, δ and β = γ, δ, since we are here composing the skew symmetric endomorphism J βαγ with the symmetric I δ . Thus, any pair in the first family {J αβ } with α < β listed in (4.1) is given by orthogonal complex structures. Similarly, one sees that any pair chosen inside the family {J αβγ }, for α < β < γ, or inside the family {J αβγδǫζ }, for α < β < γ < δ < ǫ < ζ, is given also by orthogonal complex structures.
It remains to be shown that complex structures chosen in different families among the three listed in (4.1) are mutually orthogonal. We do this for J αβ and J γδǫ and without coincidences between any of the first two and any of the last three indices (when there is at least one coincidence, the orthogonality is immediate). Remind that the composition J αβγδǫ = J αβ • J γδǫ is a self adjoint involution and, if the index ζ is distinct from all the five previous ones, J αβγδǫ anti-commutes with I ζ . Thus, we are dealing with the two self adjoint anti-commuting involutions A = J αβγδǫ and B = I ζ . Let B ′ = C −1 BC = diag = (λ 1 , . . . λ 16 ) be a diagonal form of B and let A ′ = C −1 AC. Since B ′2 = B 2 is the identity matrix, we have λ 2 1 = · · · = λ 2 16 = 1. On the other hand, from AB = −BA we get A ′ B ′ = −B ′ A ′ , so that the diagonal entries of A ′ are all zero. It follows that the trace of A = J αβγδǫ is zero, and thus tr(J * αβ J γδǫ ) = 0, showing that J αβ and J γδǫ are orthogonal. The other possibilities of two complex structures in different families listed in (4.1) are treated in a similar way.
Both definitions of a Sp(2) · Sp(1)-structure and of a Spin(9)-structure (cf. Definition 2.1 and the discussion at the very beginning of this Section) fit in the framework of the so-called Clifford systems (see [FKM81,Rad14,GR15]). These are sets C = (P 0 , . . . , P m ) of symmetric transformations in a Euclidean real vector space R N such that P 2 α = Id for all α and P α P β + P β P α = 0 for all α = β. One can then show that a Clifford system exists in a R N if and only if N = 2kδ(m), where k is a positive integer and δ(m) is given by: When k = 1 the Clifford system is said to be irreducible. Thus, for m = 8 and for m = 4, an irreducible Clifford system defines a Spin(9) and a Sp(2) · Sp(1) structure in R 16 and in R 8 , respectively. The prototype example of an irreducible Clifford system is, for m = 2 and in R 4 ≡ C 2 , the set of the three Pauli matrices The former table foresees the existence of an irreducible Clifford system with m = 9 in the vector space R 32 . Be careful that this does not contradict Proposition 4.1, stating that such a Clifford system cannot be chosen with all elements in U(16). To write a Clifford system C 9 = (P 0 , P 1 , . . . , P 9 ) in R 32 , one can imitate the procedure that allows to pass from C 4 to C 8 , i.e. from a Sp(2) · Sp(1) to a Spin(9) structure. This gives the following symmetric matrices in SO (32): where the J 1β are the ones defined in (2.8), (2.10). It is immediately checked that P 2 α = Id and P α P β = −P β P α for α = β.
The following complex structures P αβ = P α • P β (α < β) in R 32 generate a Lie subalgebra h ⊂ so(32). We will see in a moment that h ∼ = spin(10) ⊂ su(16) ⊂ so(32). The 45 P αβ can be split into the following three families of respectively 8, 28 and 9 skew-symmetric matrices: Note that by construction the vector space generated by all these P αβ is a Lie subalgebra h of so(32).

The Lie algebra spin(10) ⊂ su(16)
To relate the Lie algebra h =< P αβ > 0≤α<β≤9 constructed in the previous Section with the Lie algebra spin(10), it is useful to compare it with a description of the (half-)spin representation of the group Spin(10) on C 16 . References for spin representations are for example [Pos86, Lecture 13] and [Mei13, Chapter 3]. However, a specific (and for us convenient) excellent account to the group Spin(10) has been given by R. Bryant in the file [Bry99]. Since the representation of Spin(9) ⊂ SO(16) in Bryant's notes is slightly different from the one used in R. Harvey's book [Har90], and since we used this latter both in our previous papers [PP12,PP13,OPPV13] and in the previous Sections, we need first to rephrase in our context some arguments.
At Lie algebras level, we can go from spin(9) to spin(10) by adding to the family J C = {J αβ } 1≤α<β≤9 of 36 complex structures nine further complex structures in C 16 . Since the spin representation of Spin(10) is on C 16 , the new family will be of 45 complex structures on C 16 , and a basis of spin(10).
In the approach of [Bry99], one first looks at Spin(10) as a subgroup of Cl(R ⊕ O, <, >), the Clifford algebra generated by R ⊕ O endowed with its direct sum inner product. This algebra is isomorphic to End C (C ⊗ O 2 ): since this latter is isomorphic to M 16 (C), the linear map has to be, by dimensional reasons, a one-to-one onto representation, whence the claimed isomorphism This allows to recognize the Lie algebra spin(10) as: This description is consistent with obtaining spin(10) through the datum of the nine extra complex structures J 01 = I 0 I 1 , J 02 = I 0 I 2 , . . . , J 09 = I 0 I 9 to be added to the family J C of the 36 complex structures defining its Lie sub-algebra In particular, the inclusions spin(9) ⊂ so(16), spin(10) ⊂ su (16) are immediately recognized. Observe also that, since there are no intermediate subgroups between Spin(9) and Spin(10), the latter is generated by its subgroup Spin(9) and by the circle Moreover, looking back to the the family J C of complex structures and their Kähler forms given by (2.11), (2.13), note that once the 8 complex structures (2.13) are given, one can construct from them all the remaining 28 complex structures as: ]. Thus, a coherent way to define new complex structures J 01 , J 02 , . . . , J 09 in C 16 , and to obtain thus a basis of spin(10), is given as follows: Then, by reading Formulas (2.11) in complex coordinates, we get: The "new" Kähler forms ψ 01 , ψ 02 , . . . , ψ 09 , associated with J 01 , J 02 , . . . , J 09 , read: Then: Proposition 5.1. The Lie subalgebras h and spin(10) of so(32) are isomorphic.
Proof. An isomorphism can be defined through the choices of our bases. Just look at the correspondence Beginning of the Proof of Theorem 1.1. The two isomorphic Lie subalgebras h, spin(10) ⊂ so(32) correspond to two subgroups H, H ′ ⊂ SO(32). Note that both H and H ′ are isomorphic to Spin(10). For the subgroup H, this is recognized by the characterization of Spin(n) as the group generated by unit bivectors in the multiplicative group of invertible element in the ambient Clifford algebra (see for example [Har90,page 198]). As for H ′ , its isomorphism with Spin(10) is a consequence of how we constructed the group H ′ ⊂ SU(16) and its Lie algebra at the beginning of this Section. Note that in the previous discussion we already denoted by Spin(10) the subgroup H ′ ⊂ SU(16) and by spin(10) its Lie algebra. By comparing with the half-spin representation theory ([Mei13, Chapter 3], in particular pages 80-85), it follows that H is a real non-spin representation of Spin(10) and that H ′ is the image under one of the two non-isomorphic and conjugate half-spin representations of the abstract group Spin(10).

The canonical 8-form Φ Spin(10)
Look now at the skew-symmetric matrix ψ D = (ψ αβ ) 0≤α,β≤9 of the Kähler forms defined for α < β in the previous Section and associated with the family J D . The skew-symmetry of ψ D as matrix of Kähler forms associated with complex structures is insured by setting (formally, coherently with Proposition 5.1 and for α = 1, . . . , 9): αβ the second coefficient of its characteristic polynomial. Theorem 6.1.
is the Kähler 2-form of the complex structure I on C 16 .
It is now natural to give the following Definition 6.2. Let τ 4 be the fourth coefficient of the characteristic polynomial. We call the 8-form the canonical 8-form associated with the standard Spin(10)-structure in C 16 .
Remark 6.3. A close analogy appears between the constructions of the forms Φ Spin(9) ∈ Λ 8 (R 16 ) and Φ Spin(10) ∈ Λ 8 (C 16 ) (cf. Proposition 2.2 and Definition 6.2). However, Φ Spin(9) can alternatively be defined by integrating the volume of octonionic lines in the octonionic plane. Namely, if ν l denotes the volume form on the line l where p l : O 2 ∼ = R 16 → l is the orthogonal projection and OP 1 ∼ = S 8 is the octonionic projective line of all the lines l ⊂ O 2 . This is the definition of Φ Spin(9) proposed by M. Berger in [Ber72], somehow anticipating the spirit of calibrations. Of course, an approach like this is not possible for Φ Spin(10) , due to the lack of a similar Hopf fibration to refer to. Thus Definition 6.2 appears to be a coherent algebraic analogy, and as we will see in next Section, it is suitable to represent a generator for the cohomology of the relevant symmetric space.
Remark 6.4. Denote by I the standard complex structure on C 16 and look at the ten endomorphisms I, I 1 , . . . , I 9 : the first of them is a complex structure and the remaining nine are involutions. The above discussion shows that these data are the right choice to give rise, via compositions of any pair of the ten endomorphisms, to the family J D = {J αβ } 0≤α<β≤9 , a basis of spin(10). Note also that, on the fourth Severi variety EIII, the complex structure I can be looked at as element of the Lie algebra in the second factor of its holonomy Spin(10) · U(1). 7. The even Clifford structure, cohomology and proof of Theorem 1.2 In [MS11] the notion of even Clifford structure on a Riemannian manifold (M, g) is proposed as the datum of an oriented rank r Euclidean vector bundle E → M , together with a bundle morphism ϕ : Cl 0 (E) → End(T M ) from the even Clifford algebra bundle of E, and mapping Λ 2 E into the skew-symmetric endomorphisms. Here Λ 2 E is viewed as a sub-bundle of Cl 0 (E) by the identification e ∧ f ∼ e · f + h(e, f ), for each e, f ∈ E and where h is the Euclidean metric on E.
Under the hypothesis of parallel even Clifford structure (cf. [MS11, page 945]), complete simply connected Riemannian manifolds admitting such a structure are classified ([MS11, page 955]) and EIII turns out to be the only non-flat example with r = 10.
The (rational) cohomology of the fourth Severi variety V (4) ∼ = EIII can be computed from the so-called A. Borel presentation. Let G be a compact connected Lie group, let H be a closed connected subgroup of maximal rank and let T be a common maximal torus. Then (cf. [Bor53, §26, page 19]): where BT is the classifying space of the torus T , H * (BT ) W (H) is the invariant sub-algebra of the Weyl group W (H), and H >0 (BT ) W (G) denotes the component in positive degree of the invariant sub-algebra of the Weyl group W (G). One gets in this way the cohomology structure given by (3.2). Thus: Corollary 7.2. The Poincaré polynomial, the Euler characteristics and the signature of EIII are given by Poin EIII = 1 + t 2 + t 4 + t 6 + 2t 8 + 2t 10 + 2t 12 + 2t 14 + 3t 16 + . . . , Since EIII can be looked at as the projective plane over the complex octonions, it is natural to similarly construct a projective line over complex octonions, that turns out to be a totally geodesic submanifold of the former [Esc12a,Esc12b]. This is the oriented Grassmannian: which is a non singular quadric Q 8 ⊂ CP 9 , thus again a Hermitian symmetric space. Its rational cohomology is given by: where e ∈ H 2 and e ⊥ ∈ H 8 are the Euler classes of the tautological vector bundle and of its orthogonal complement, and the relations are: ρ 5 = ee ⊥ ∈ H 10 , ρ 8 = e 8 − (e ⊥ ) 2 ∈ H 16 . Thus: Corollary 7.4. The Poincaré polynomial, the Euler characteristics and signature are
We need now the following fact. The Kähler 2-forms ψ αβ (0 ≤ α < β ≤ 9), that we wrote explicitly and globally on C 16 , are of course only local on EIII. They are associated with its non-flat even parallel rank 10 Clifford structure. In situations like this it has been proved that such Kähler 2-forms turn out to be proportional to the curvature forms Ω αβ of a metric connection on the structure bundle. An observation like this can be traced back to S. Ishihara [Ish74] in the context of quaternion-Kähler manifolds, where the local Kähler 2-forms associated with the local compatible almost complex structures I, J, K are recognized to be proportional to the curvature forms. Later, a similar argument has been developed by A. Moroianu and U. Semmelmann to get the same identification on Riemannian manifolds M equipped with a non-flat parallel even Clifford structure [MS11, Prop. 2.10 (ii) (a) at page 947 and Formula (14) at page 949]. In all these contexts, the Einstein property of the manifold is insured from the hypotheses. The proportionality stated in [MS11] reads: where κ is deduced from the Ricci endomorphism Ric on M as follows Ric = κ(n/4 + 2r − 4), and n, r are the real dimension of the manifold and the rank of its non-flat even Clifford structure, respectively. Note that this insures that the 8-form τ 4 (ψ D ) on EIII is closed, ending the proof of Theorem 1.1.
Coming now to Theorem 1.2, look at the totally geodesic Q 8 ∼ = Gr 2 (R 10 ). On Q 8 the restriction of our ψ αβ , (1 ≤ α < β ≤ 8) defines a rank 8 even Clifford structure. We normalize the metric on EIII in such a way that the induced metric on Q 8 is the same as the one induced by Q 8 ⊂ CP 9 , with CP 9 of holomorphic sectional curvature 4. This choice gives Ric(CP 9 ) = 20 and Ric(Q 8 ) = 16, so that the above identity with n = 16 and r = 8 gives κ = 1. Therefore on the quadric Q 8 : Thus the ψ αβ are local curvature forms of a metric connection on the rank 8 Euclidean vector bundle over Gr 2 (R 10 ) having (7.6) as associated sphere bundle. This vector bundle is easily recognized to be γ ⊥ 2 (R 10 ), the orthogonal complement in R 10 of the tautological plane bundle over Gr 2 (R 10 ), and it defines a non-flat even rank 8 Clifford structure on Gr 2 (R 10 ), cf. [MS11, Tables at pages 955 and 965]. Using Chern-Weil theory, and recalling that ψ 09 | C 8 = i 2 11 + · · · + 88 and ψ 0β | C 8 = 0 otherwise, we get: (7.7) τ 4 ψ D | Gr2(R 10 ) = (2π) 4 p 2 (γ ⊥ 2 (R 10 )) − 4ω 4 . Here p 2 is the second Pontrjagin class and the last coefficient −4 comes from: cf. beginning of the proof in 6.1.
The equality 7.7 can be reread by restricting at the maximal linear subspaces CP 4 , (CP 4 ) ′ , that parametrize those oriented 2-planes in R 10 that are complex lines with respect to a complex structure preserving or reversing the orientation, cf. Remark 3.1. If p 2 = p 2 (γ ⊥ 1 (C 5 )) is now the second Pontrjagin class of the orthogonal complement of the tautological line bundle over CP 4 , this gives: 1 (2π) 4 On the other hand the Pontrjagin class p 2 (γ ⊥ 1 (C 5 )) relates with the Chern classes of the same bundle as p 2 = 2c 4 − 2c 1 c 3 + c 2 2 , giving again the generator of the top integral cohomology class. Thus, taking into account the orientation matter concerning CP 4 and (CP 4 ) ′ and discussed in Remark 3.1, one gets: