3. Self maps of Homogeneous spaces

This work (done jointly with V. Srinivas) arose out of an attempt to understand the following problem of Lazarsfeld [14]:

Problem 3.1. Suppose G is a semi-simple algebraic group over C, P G a maximal parabolic subgroup, Y = G∕P. Let f : Y X be a finite, surjective morphism of degree > 1 to a smooth variety X; then is X~=Pn ? (n = dimX = dimY )

Lazarsfeld (loc. cit.) answers this in the affirmative when Y = Pn, using the proof by S. Mori [16] of Hartshorne’s conjecture. The general case seems to be open even for Grassmann varieties.

We show that:

if Y = G∕P is as above and f : Y Y is a finite self map of degree > 1, then Y ~=Pn.

More generally, we prove the following:

Theorem 3.2. Let G be a simply connected, semi-simple algebraic group over C. Let P G be a parabolic subgroup, and let Y = G∕P be the homogeneous space. Let f : Y Y be a generically finite morphism. Then there exist parabolic subgroups P0,P1,,Pm of G containing P, and a permutation σ of {1,2,,m} such that:

  1. there are isomorphisms G∕Pi~=Pni for i 1, for some integers ni > 0, such that nσ(i) = ni for all i.
  2. there is a finite morphism πi : Pni Pni for each i > 0.
  3. the natural morphism
    Y →  G∕P0 × G ∕P1 × ⋅⋅⋅× G∕Pm

    is an isomorphism, under which f : Y Y corresponds to the product f0 × f1 ×⋅⋅⋅× fm, where f0 : G∕P0 G∕P0 is an isomorphism and fi the composite

            ~   ni πi ni ~
G ∕Pσ(i)= P   →  P  =  G∕Pi.

Next, we prove

Theorem 3.3. Problem 1 has an affirmative answer if Y is a smooth quadric hypersurface of dimension 3.

Remark: This includes the case of the Grassmannian Y = G(2,4).

We also show:

Proposition 3.4. Let k n,2 l m be integers, such that there exists a finite surjective morphism between Grassmann varieties

f : G(k,k + n) → G (l,l + m ).

Then k = l,m = n and f is an isomorphism.

In contrast with the above algebraic results, we can construct topological maps of degree greater than one in the following cases:

  1. If Y is a smooth quadric hypersurface of odd dimension n we have maps Y Pn.
  2. If Y is a smooth quadric hypersurface of any dimension we have maps Y Y .