Morphisms between Grassmannians, II

Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that, if $\varphi:\mathbb G(l,n) \to \mathbb G(k,n)$ is a non constant morphism and $l \not=0,n-1$ then $l=k$ or $l=n-k-1$ and $\varphi$ is an isomorphism.


Introduction
In [10], Tango proved that there are no nonconstant morphisms from P m to the Grassmannian G(k, n) if m > n, and later, in [11], considered the case m = n, proving the same result for kn even, (k, n) = (2, 5), k ∈ {0, n − 1}.The result in [10] has been generalized in [8] to the case of morphisms ϕ : G(l, m) → G(k, n), with m > n, and later to a more general setting (see Theorem 2.3 and references therein).
The aim of the present paper is to generalize the results in [11], considering morphisms G(l, n) → G(k, n), and proving the following: Theorem 1.1.If ϕ : G(l, n) → G(k, n) is a nonconstant morphism and l = 0, n− 1 then l = k or l = n − k − 1 and ϕ is an isomorphism.
In a nutshell, the idea in [10] and [8] to prove the constancy of a morphism ϕ : M → G(k, n) was to consider the relation among Chern classes coming from the universal sequence on G(k, n), pull it back via ϕ, and show that this leads to a contradiction, via a study of the Chow ring A • (M ).
In [7] this idea was refined and reinterpreted geometrically, considering the Schubert varieties X H , X p ⊂ G(k, n), parametrizing linear spaces contained in a hyperplane H or passing through a point p; then [X H ] ∈ A k+1 (G(k, n)), [X p ] ∈ A n−k (G(k, n)) and clearly [X H ] • [X p ] = 0.If the pullback via ϕ of one of the two cycles is zero one can construct a morphism from M to a smaller Grassmannian which factors via ϕ, allowing inductive arguments.Else, one obtains two effective nonzero cycles whose product is zero in A n+1 (M ).The proof can then be finished by showing that there are no such pairs in A • (M ).This is the idea that led to the notion of effective good divisibility of a variety (see Section 2.1).
The last step of the above argument could be further refined: in fact it is enough to show that for every effective nonzero x ∈ A k+1 (M ), y ∈ A n−k (M ) it holds x • y = 0. We fulfill this task, in our setup, by characterizing the pairs of effective nonzero cycles of total codimension n + 1 in A • (G(l, n)) whose product is zero, called maximal disjoint pairs (Corollary 3.3).In particular, for G(l, n) such a pair consists of cycles of codimensions l + 1 and n − l, forcing l = k or n − k − 1 in order for ϕ : G(l, n) → G(k, n) to be nonconstant.In these cases we conclude that ϕ is an isomorphism using a Remmert-Van de Ven type theorem for rational homogeneous varieties due to Hwang and Mok ([6, Main Theorem]).

Preliminaries
2.1.Effective Good Divisibility.The notion of effective good divisibility of a smooth complex projective variety M (see [7,Section 2.1]) is related to the total codimension of effective zero divisors in the Chow ring A • (M ).
Definition 2.1.The effective good divisibility of M , denoted by e.d.(M ), is the maximum integer s such that, given effective cycles x i ∈ A i (M ), x j ∈ A j (M ) with i + j ≤ s and x i x j = 0, then either x i = 0 or x j = 0.
Example 2.2.The effective good divisibility is known for the wide class of rational homogeneous manifolds: it was computed for Grassmannians in [8], for varieties of classical type independently in [7] and [5], for varieties of exceptional type in [5].In this paper we are mostly interested in Grassmannians G(k, n), parametrizing linear subspaces of dimension k in P n , for which e.d.(G(k, n)) = n.
Knowledge of the effective good divisibility can be used to prove the non existence of nonconstant morphisms, as exemplified by the following result.
In order to prove the non existence of nonconstant morphisms between two Grassmannians G(l, n) and G(k, n), which have the same effective good divisibility we need to characterize the pairs of effective zero divisors (x i , x j ) of minimal total codimension i + j; they have been introduced in [7, Definition 2.3].In the definition given below we consider also the type of the pair, which keeps track of the codimensions of x i and x j .Definition 2.4.A (non-ordered) pair {x i , x j } with x i ∈ A i (M ), x j ∈ A j (M ) nonzero effective cycles such that x i x j = 0 and i + j = e.d.(M ) + 1 will be called a maximal disjoint pair (md-pair for short) of type {i, j} in M .
We will prove Theorem 1.1 using the fact that two non-isomorphic Grassmannians G(l, n) and G(k, n) do not possess md-pairs of the same type, so we review in the next subsection basic facts about the generators of the Chow ring A • (G(k, n)).

Schubert varieties.
We will recall some basic facts about Schubert calculus in G(k, n).We refer to [1] and [3] for the proofs and for a complete account on the subject.
Let us identify G(k, n) with the Grassmannian G(k + 1, V ) of vector subspaces of dimension k + 1 in a vector space V of dimension n + 1 and consider a complete flag V of vectors subspaces of V :

Given a sequence of integers
To the Schubert variety X I one can associate a Young diagram λ I in the following way: consider a rectangle with k + 1 rows and n − k columns and the path from the lower-left corner to the upper-right corner consisting of n + 1 steps, where the i-th step is vertical if i ∈ I and horizontal if i ∈ I.The Young diagram is the part of the rectangle that is top-left of this path.
Identifying the Young diagram λ I with the partition (λ 1 , . . ., λ k+1 ), where λ i is the number of boxes in row i, we have: By abuse, we denote by λ I also the corresponding partition (λ 1 , . . ., λ k+1 ), and we set |I| := |λ I | = λ i ; this number is the dimension of the subvariety X I .
The Bruhat order on the set of Schubert symbols is defined as follows: I ≤ L if and only if i j ≤ l j for every j = 1, . . ., n + 1.Notice that does not depend on the choice of the flag V, and will be called a Schubert cycle.The Schubert cycles form a basis of A Let p = P(V 1 ); the Schubert variety X p := X Ip , corresponding to the Schubert symbol A dual description of Schubert varieties, which highlights the codimension of the variety rather than the dimension, is possible.Given a Schubert symbol In particular the diagram λ I ∨ is obtained from the diagram λ I by taking the complement and rotating it by 180 Example 2.6.Let I p and I H be the Schubert symbols introduced in Example 2.5.

Then the Young diagram of λ I ∨
H corresponds to the partition (1, . . ., 1), while the one of λ I ∨ p corresponds to the partition (n − k, 0, . . ., 0).Again, we show the diagrams for the case n = 6, k = 2.
Given a Schubert index I we define the opposite Schubert variety X I to be w 0 X I ∨ , where w 0 is the longest element of the Weyl group of GL(n + 1, C), which acts on the canonical basis as w 0 (e i ) = e n+1−i .Clearly we have an equality of cycles [ We are going to use the following well-known fact [2, Section 1. 3

Md-pairs in Grassmannians
In this section we will show that the only md-pair for G(k, n), 1 ≤ k ≤ n − 2 is essentially the one described in Example 2.5; let us start by considering md-pairs whose elements are Schubert cycles.Theorem 3.1.In the Grassmannian G(k, n), with 1 ≤ k ≤ n − 2, let X I , X J be Schubert varieties with codim(X I ) + codim(X J ) ≤ n + 1.Then [X I ] • [X J ] = 0 if and only if {I, J} = {I H , I p }.
Proof.Since [X I ] = [X I ∨ ], by Proposition 2.7 we know that [X I ] • [X J ] = 0 if and only if I ∨ ≤ J.This in turn happens if and only if the Young diagram of I ∨ is contained in the Young diagram of J, i.e., the associated partitions λ I ∨ and µ J satisfy λ ∨ i ≤ µ i for 1 ≤ i ≤ k + 1.Note that the assumption on the codimensions can be rewritten as which can be restated as (3) |λ ) or ((1, . . ., 1), (n − k, . . ., n − k, 0)) and from Example 2.5 we obtain the statement.
Using the inequalities (4) we see that hence, using (5) we obtain forcing m = 1 and λ j = 0 for j ≥ 2.Moreover, all inequalities in ( 6) and ( 7) are equalities.In particular Note that here we are using the assumption k ≥ 1).
We can now describe the md-pairs of G(k, n).[Γ] = ] is a combination of Schubert cycles with nonnegative coefficients, due to the Littlewood-Richardson rule.Then, if γ K δ L = 0 we must have [X K ] • [X L ] = 0, and the statement now follows from Theorem 3.1.

Morphisms to Grassmannians
In this section we will prove Theorem 1.1.We state and prove first two auxiliary results that could be useful to study morphisms to Grassmannians from other kinds of varieties.
Proof.In the first case, for a general hyperplane H ′ , Y does not meet the subvariety X H ′ ⊂ G(k, n) parametrizing linear spaces contained in H ′ .We can take ψ to be the restriction to Y of the morphism π H : G(k, n) \ X H ′ → G(k − 1, n − 1), which sends Λ to Λ ∩ H ′ .Since the fibers of this morphism are affine (see [7, Example 5.5]), we get that ψ is not constant.
The argument in the second case is similar: for a general point q ∈ P n , Y does not meet the subvariety X q ⊂ G(k, n) parametrizing linear spaces passing through q.We can take ψ to be the restriction to Y of the morphism π q : G(k, n) \ X q → G(k, n − 1), which sends Λ to the linear projection of Λ from q to a hyperplane.Again the fibers of this morphism are affine and ψ is not constant.
be as in Example 2.5.For general g, g ′ ∈ GL(n + 1, C), gX H and g ′ X p are disjoint and generically transverse to ϕ, that is, ϕ −1 (gX H ), ϕ −1 (g ′ X p ) are generically reduced and of the same codimensions as X H , X p ⊂ X (see [1,Theorem 1.7]).
By Proof of Theorem 1.1.Since the Picard number of G(l, n) is one, a nonconstant morphism must be finite.In fact, if this were not the case, the pullback of an ample line bundle on the image will be ample on G(l, n), but trivial on the contracted curves, contradicting Kleiman's criterion.If k = 0, n − 1 we then have a morphism ϕ : G(l, n) → P n which is constant since dim G(l, n) > n.Else, by Proposition 4.2, if ϕ is not constant then l = k or n − k − 1.In this case, since the dimensions of the domain and the codomain are equal, ϕ must be surjective.We conclude that ϕ is an isomorphism by [6,Main Theorem].
or equivalently k-dimensional linear subspaces of P(V ) containing p.The associated Young diagram λ Ip is the diagram whose partition is (n − k, . . ., n − k, 0).Let us show the diagrams λ IH and λ Ip for n = 6, k = 2.

Proposition 4 . 2 .
Let k, n be integers such that 1 ≤ k ≤ n − 2 and let M be a smooth variety with e.d.(M ) = n in which there is no md-pair of type {k + 1, n − k}.Then every morphism ϕ :
Proposition 2.7.The intersection of a Schubert variety X J and an opposite Schubert variety X I is nonempty if and only if I ≤ J.This is the case if and only if the intersection product[X I ] • [X J ] is nonzero.Example 2.8.Let I H and I p be as in Example 2.5.Clearly, recalling the geometric descriptions of X IH and X Ip , we have that [X IH ] • [X Ip ] = 0, but we can obtain this also using Proposition 2.7, since I ∨ H ≤ I p (and dually I ∨ p ≤ I H ).