In [Math. Ann. 142, 453-468], Remmert and Van de Ven conjectured that if
X
X
is the image of a surjective holomorphic map from
P
n
\mathbb {P}^n
, then
X
X
is biholomorphic to
P
n
\mathbb {P}^n
. This conjecture was proved by Lazarsfeld [Lect. Notes Math. 1092 (1984), 29-61] using Mori’s proof of Hartshorne’s conjecture [Ann. Math. 110 (1979), 593-606]. Then Lazarsfeld raised a more general problem, which was completely answered in the positive by Hwang and Mok.
Theorem 1 ([Invent. math. 136 (1999), 209-231] and [Asian J. Math. 8 (2004), 51-63]). Let
S
=
G
/
P
S=G/P
be a rational homogeneous manifold of Picard number
1
1
. For any surjective holomorphic map
f
:
S
→
X
f:S\to X
to a projective manifold
X
X
, either
X
X
is a projective space, or
f
f
is a biholomorphism.
The aim of this article is to give a generalization of Theorem 1. We will show that modulo canonical projections, Theorem 1 is true when
G
G
is simple without the assumption on Picard number. We need to find a dominating and generically unsplit family of rational curves which are of positive degree with respect to a given nef line bundle on
X
X
. Such a family may not exist in general, but we will prove its existence under a certain assumption which is applicable in our situation.