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 XPⁿ ? (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.

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

Pⁿ.

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,…,P_m of G containing P, and a permutation σ of {1,2,…,m} such that:

there are isomorphisms G∕P_iP^n_i for i ≥ 1, for some integers n_i > 0, such that n_σ(i) = n_i for all i.
there is a finite morphism π_i : P^n_i → P^n_i for each i > 0.
the natural morphism $Y → G∕P0 × G ∕P1 × ⋅⋅⋅× G∕Pm$
is an isomorphism, under which f : Y → Y corresponds to the product f₀ × f₁ ×× f_m, where f₀ : G∕P₀ → G∕P₀ is an isomorphism and f_i the composite
$~ ni πi ni ~ G ∕Pσ(i)= P → P = G∕Pi.$

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

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:

3. Self maps of Homogeneous spaces