2025-08-26

Setup

Consider a vector bundle :

Pr¡1EP(E)XXiÃvectorbundlerankr

We can projectivise to , and has a canonical line bundle given by each point in corresponding to a line . We have

By Leray-Hirsch, there is a surjection

There is a spectral sequence

which degenerates on because of the surjection above (the vertical line must all stay, then compute via the Leibniz rule). Now, is a -module with basis , where is the generator of . In particular,

where are the Chern classes.

Exercise

Let and . We have

where this really means

where are the Chern classes of .

Remark

This works for Grassmannian bundles as well. For example, if , we get

where .

The exercise implies that

Recall that we had

  • . We can the replace the equation with .
  • …
  • Continue to get finally same thing but with equation .

Other types

We have

This is not true integrally, but we only need to invert the worst primes to make it true.

Example:

Siegel parabolic:

Definition (Parabolic)

connected?

A parabolic subgroup is a subgroup that contains a Borel. The following are equivalent:

  • is parabolic.
  • is proper.
  • is projective.
  • There exists such that

Exercise

Find that gives the Siegel parabolic.

More on

We have

Notice that . We have

So we have

  • is given by an intersection with a hyperplane.
  • It is a quadric hypersurface in .

.
A symplectic form is a form on .

:

Fact: Given a smooth degree hypersurface in :

Intersecting a hypersurface with generic hyperplanes gives points.

This implies that is not generated by . Also true the full flag variety: is not generated by .

Schubert varieties

Let . Bruhat: . Let and .

  • a Schubert cell. Topologically, these are all .
  • a Schubert variety.

Theorem

is a basis of .

Exercise

In type A, Schubert varieties have the following descriptions: Fix a flag

stable under . Then

where depend on .

Remark

is often singular.

Example

Consider :

  • Longest root .
  • Two reflections and .

:

Look at action of . It connects and by a in .

Implies . is a -submodule. Fact: generates as a -module ( is highest root, so is smallest possible root). Now this implies . Now but , not smooth. .