Vector bundle

We will use the Weil conjectures to compute the Betti numbers of Hilbert schemes. They have been used before to compute Betti numbers of algebraic varieties, at least since in [Harder-Narasimhan (1)] they were applied for moduli spaces of vector bundles on smooth curves. 

Notation. In chapter 3 and 4 we will often use the Grassmannian bundle associated to a vector bundle. So we fix some notations for these. 

For the Chern classes of a symmetric power of a vector bundle we have the well-known relation ... 

From the above we see that for a vector bundle of rank r over X with Chern classes ei,..., er the formula... 

We now want to show that D (X) and D 1 X) are again Grassmannian bundles corresponding to vector bundles over D X) and D j l(X) respectively. Before doing this we want to show that these two cases are the only ones in which we can expect such a result (exept for the trivial case m — d). 

Let be a vector bundle of rank r on a smooth projective variety X. Now we want to study the vector bundles (P) from definition 3.2.6. For this purpose we first consider the bundles Ei on the Hilbert scheme X. We can associate in a natural way to each section sof a section sj of Ei and thus also a section (s)JJ, of... 

It is well known that the relative tangent bundle of a projectivized vector bundle E of rank r is... 

As the Chern classes of symmetric powers of vector bundles of rank 2 axe easy to compute, we know now the Chow ring of Hilbn(P( )/A). In particular we obtain ... 

Corollary 4.1.12. If E is a vector bundle of rank 2 over X with Chern classes ci,C2, then... 

Now we want to generalize the result of the last section. Let X be a smooth variety and E a vector bundle of rank 3 on X. 

As in section 4.3 let E be a vector bundle of rank 3 over a smooth variety X. We... 

On the Cohomology Groups of Moduli Spaces of Vector Bundles on Curves, Math. Ann. 212 (1975), 215-248. 

An algebraic fibre bundle over Pi, that is not a vector bundle, Topology 12 (1973), 229-232. 

Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127 (1988), 309-316. 

Cycles, curves and vector bundles on an algebraic surface, Duke Math. J. 54 (1987), 1-26. 

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