Introduction

Let G be a simply connected, semi-simple algebraic group over $\mbox{$\bf C$}$. Let P $\subset$ G be a parabolic subgroup, and let Y = G/P be the homogeneous space. In a recent paper [KS], we showed that

if Y = G/P is as above and f : Y$\to$Y is a finite (algebraic) self map of degree > 1, then Y $\cong$ $\mbox{$\bf P$}$ⁿ.

The paper arose out of an attempt to understand the following problem of Lazarsfeld [L]:

Problem -99   Suppose G is a semi-simple algebraic group over $\bf C$, P $\subset$ G a maximal parabolic subgroup, Y = G/P. Let f : Y$\to$X be a finite, surjective morphism of degree > 1 to a smooth variety X; then is X $\cong$ $\mbox{$\bf P$}$ⁿ? ( n = dim X = dim Y)

Lazarsfeld (loc. cit.) answers this in the affirmative when Y = $\mbox{$\bf P$}$ⁿ, using the proof by S. Mori [M] of Hartshorne's conjecture. We also showed that Lazarsfeld's problem has an affirmative answer if Y is a smooth quadric hypersurface of dimension $\geq$ 3. This includes the case of the Grassmannian Y = $\mbox{$\bf G$}$(2, 4). The general case seems to be open even for other Grassmann varieties. Our goal in this paper is to study the analogous problems for continuous maps. Our homogeneous spaces are all complex submanifolds of complex projective spaces $\mbox{$\bf C$}$$\mbox{$\bf P$}$ⁿ, with the usual topology; we drop the $\mbox{$\bf C$}$ to simplify notation. We show

Theorem 1   Let Q be a smooth quadric hypersurface in $\mbox{$\bf P$}$^{n + 1}, where n = 2k + 1. Then for any positive integer d $\equiv$ 0  (mod 2^k) there exist continuous maps f : $\mbox{$\bf P$}$ⁿ$\to$Q satisfying f^*($\mbox{${\cal O}_{Q}$}$(1)) = $\mbox{${\cal O}_{\P^n}$}$(d ).

Note that such maps have degree greater than one whenever d > 1. We also show

Theorem 2   Let Q be a smooth quadric hypersurface in $\mbox{$\bf P$}$^{n + 1}. Then there exists a positive integer m and continuous maps of degree (m ^. d )ⁿ from Q to Q, for all d $\in$ $\mbox{$\bf N$}$.

Clearly Theorem 1 implies Theorem 2 in the case when n is odd, since there is an obvious map Q$\to$$\mbox{$\bf P$}$ⁿ of degree 2. However, we also construct self maps of Q of odd degree, when n is odd. Observe that the degree of any self map of Q is an n-th power. Our proofs are by obstruction theory, using the standard (Bruhat) cell decomposition, and by induction on the dimension. The cases when n is even and odd are dealt with separately, and the induction is in steps of 2. From the computation of the homotopy groups of quadrics, such a division seems natural. In the paper [KS], we had also proved:

Proposition 0   Let k $\leq$ n, 2 $\leq$ l $\leq$ m be integers, such that there exists a finite surjective morphism between Grassmann varieties

f : $$\mbox{$\bf G$}$$(k, k + n)$$\to$$$$\mbox{$\bf G$}$$(l, l + m).

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

We do not know whether there are continuous maps of degree bigger than one f : $\mbox{$\bf G$}$(k, k + n)$\to$$\mbox{$\bf G$}$(l, l + m) with k $\neq$ l, m $\neq$ n and kn = lm. We wish to thank P. Polo and A. R. Shastri for stimulating discussions.