Bundle isomorphism

The notion of bundle isomorphism generalizes straightforwardly by requiring that the isomorphism map be T -invariant. To summarize, we have a category of principal F-bundles as objects and principal T-bundle maps as morphisms. 

Obviously Z(12)(Jf) is isomorphic to the Grassmannian bundle Grass(2,Tfr) of two-dimensional quotients of the cotangent bundle of X. We put... 

Because the bundles have the same rank, it is an isomorphism. So (2) follows. ... 

One of the advantages of the hyper-Kahler structure is that one can identify two apparently different complex manifolds with one hyper-Kahler manifold. Namely, a hyper-Kahler manifold X, g, I, J, K) gives two complex manifolds (X,/) and (X, J), which are not isomorphic in general. For example, on a compact Riemann surface, the moduli space of Higgs bundles and the moduli space of flat PGLr(C)-bundles come from one hyper-Kahler manifold, namely moduli space of 2D-self-duality equation (see [36] for detail.)... 

Remark 7.8. In order to compute the Poincare polynomial, we do not actually need the holomorphic symplectic form on T E. The fact that it is locally isomorphic to T C is enough. Hence the above argument holds and shows Gottsche s formula also for the case of the total space of a holomorphic line bundle over E, not necessarily T E. The only difference is that the unstable manifold W becomes Lagrangian in the case of T E. 

The first two chapters are about moduli of abelian varieties, i.e. about classifying abelian varieties. An abelian variety is a proper variety endowed with a group structure. It turns out that any abelian variety A is projective there exists some ample line bundle on A. An ample line bundle determines an isogeny A A — A, which depends on C only up to algebraic equivalence. Such a morphism A is called a polarization, it is called a principal polarization if A is an isomorphism. 

We then see that non-isomorphic line bundles on a curve can embed it with isomorphic normal bundles. 

Thus a vector bundle is just a scheme over X locally isomorphic to Ar x X. The simplest way to define a vector bundle, without distinguishing one atlas , is to simply add the extra condition that the atlas is maximal, i.e., every possible open V C X and isomorphism 7r-1 (V) Ar x V compatible with the given... 

This induces an -module structure in v - The reader can check that the compatibility demanded between fa and fa over Ui fl Uj is exactly what is needed to insure that the 2 -module structures that we get on u.ru. are the same. The main theorem in this direction is that every locally free ox-module arises as the sheaf of sections of a unique vector bundle (up to isomorphism). 

In view of Lemma 2.11 it is sufficient to show that the embedding P( ) — P( (9) — X is an A -weak equivalence. But fix)m [14, 8] we know that this embedding is isomorphic to the zero section embedding of P( ) into the total space of the canonical vector bundle of rank one over P( ). The proposition then follows from 2.2. 

Theorem 2.23. — Let i Z K be a cbsed embedding of smooth schemes over S. Denote by Nx,z Z the normal vector bundle to i. Then there is a canonical isomorphism in 3, (S) of the form... 

Proof — To prove that the first condition implies the second, what we have to show is that if S is henselian local and E —S is an etale principal G-bundle over S then the morphism ((E x T)/G)et S splits. In order to find such a sphtting it is sufficient to find a G-equivariant morphism E F. Since B(F, G), is isomorphic to B,(G in. f Sm/S)jf) Proposition 1.15 implies that there exists a cartesian square of the form... 

Assume now that the second condition holds. First observe that to prove is an isomorphism in Si0f Sm/S)xts) it is sufficient to show that for any etale simplicial sheaf 3T and an etale principal G-bundle E, 9 there exists a we 3S" in the Nisnevich topology and a G-equivariant morphism from E =, i x E to E(F). Indeed, this implies that there is a section j to 3> in but this fact together with lemma 1.19 does imply formally that is an isomorphism in (Sot/S)jvis). 

A very important and frequently studied type of bundles is that of the vector bundles. In this case, the fiber is required to have the structure of a vector space (usually over the real or complex numbers), and the homeomorphism = U X F is required to be a vector space isomorphism for each x U. 

An important special case of Definition 8.6 arises when Bi = B2 and fg = ids. We can therefore speak of isomorphism classes of bundles over a fixed base space. 

Proof. Assume that a = E,B,p) is a fiber bundle, and assume that B is contractible. Let q B B he map that takes the whole space B to some point b B. By our assumptions, the maps q and idg are homotopic. It follows by Theorem 8.8 that the pullbacks q a and idgo are isomorphic. On the other hand, we see that (7 0 is a trivial brmdle, and idgo = E. ... 

Let X be a CW complex. The function P from the set of homotopy classes of maps from X to 3P, [X, BF], to the set of isomorphism classes of principal P-bundles over X that takes each continuous map f X BF to the pullback of the universal bundle over BF along f is a hijection. 

Proof. To start with, P is well-defined, since by Theorem 8.8 it takes homotopic maps f,g X —> BF to isomorphic principal F-bundles. 

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