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120 XVI DIFFERENTIAL MANIFOLDS
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PROBLEMS
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1. For each differential manifold B, let I or I(B) denote the trivial real line-bundle B x R;
it is an identity element for the tensor product of vector bundles over B. Likewise Ic or IC(B) denotes the trivial complex line-bundle B x C. The sum ml (resp. mlc) of m copies of I (resp. Ic) may be identified with the trivial bundle B x Rm (resp. B x Cm).
(a) Consider the Grassmannian Gn, p = Gn, P(R) and the subset Un, p = Un, P(R) of the
trivial bundle «I(Gn, „) = Gn, p x R" consisting of pairs (V, x) where V is a vector sub- space of dimension/? in R", and x e V. Show that Un, p is a vector subbundle of/zI(Gn, p), isomorphic to the bundle associated to the principal bundle Lnf p (16.14.2) with structure group GL(p, R), with fiber-type Rp (for the canonical action of GL(/?, R) on R" on the left). Define in the same way the complex vector bundle Un, P(C) over Gn, P(C). The bundle Un, P(R) (resp. Un, P(C)) is called the canonical (or tautological) vector bundle over Gn,p(R)(resp.Gfl,p(C)).'
In particular, when p = 1 (so that Gn, i(R) = P^i(R), Gn, i(C) = P^^C)), we write
Ln_j(l) or La_i.R(l) in place of Un,i(R), and Ln»i,c(0 in place of Un, ^C). The principal bundle P(DA(R)) (resp. P(Di(C))) (Section 16.14, Problem 5) may be identified with the complement of the zero section in Lw_i(l) (resp. Ln_ltc(l)). We denote by Lrt_i(fc) (resp. Ln_i,c(&)), for each integer k > 0, the tensor product of k copies of 1^(1) (resp. L»-lf c(l)).
(b) If 12 ^ 1, the bundles Ln(2) are trivializable as real-analytic bundles, but Ln(l) is
not trivializable as a differential bundle. (Argue by contradiction, by lifting a section which is 7*0 at each point of the space Rw+1 — {0}; show that, for each x e Pn(R), there exists a real-analytic section of Ln(l) over the whole of Pn(R) which is 7*0 at the point x.) On the other hand, the holomorphic bundles Ln, c(k) for k £> 1 are pairwise non- isomorphic and admit no holomorphic section over Pn(C) other than the zero section (same method, using Section 9.10, Problem 5). However, for each x e Pn(C) there exists a real-analytic section of LM, c(0 over Pn(C) which does not vanish at x.
(c) If we endow Rn with the Euclidean scalar product, then the mapping which sends
each p-dimensional subspace V c Rn to its orthogonal supplement V-1, of dimension n •— p, is a real-analytic isomorphism w of Gn, p onto Gn, n_p. Show that the direct sum Un, p © o»*(Un, n-p) is a trivializable bundle over Gn, p. Define in the same way a holo- morphic isomorphism w of Gn, P(C) onto Gn?n_p(C), but show that the analogous as- sertion about the canonical bundles is false (consider the isotropic vector subspaces of C"). On the other hand, there exists a rea/-analytic isomorphism o>0 of Gn, P(C) onto Gn, n-p(C) such that Un, P(C) © co£(Un, n_p(C)) is trivializable as a real-analytic bundle.
2. (a) Let L be a real or complex line-bundle over a differential manifold B. Show that
the tensor product L (x) L* is trivializable (cf. (16.18.3.5)). Likewise for real- or complex- analytic line-bundles over a real-analytic manifold, and for holomorphic line-bundles over a complex manifold. By reason of this fact, vector bundles of rank 1 are also called invertible vector bundles; we write L®*-" = L*, L®(~fc) = (L*)®fc for all k > 0, with the convention that L®° = L
(b) The tensor product of k copies of Ln(l)* (resp. Ln,c(0*) (Problem 1) is denoted b>
Ln(—k) (resp. Ln<c(—k)). Show that for each k J> 1 and each jc e Pn(C) there exists a holomorphic section of LntC(—k) over Pn(C) which does not vanish at x. |
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