A Characterization of Complex Projective Spaces by Sections of Line Bundles

Let M be a n-dimensional compact irreducible complex space with a line bundle L. It is shown that if M is completely intersected with respect to L and dimH0(M, L) = n + 1, then M is biholomorphic to a complex projective space Pn of dimension n.


Introduction
Kobayashi and Ochiai [1] have given Characterizations of the complex projective spaces.Kobayashi-Ochiai Theorem [1] has been applied to obtain many important characterizations of the projective spaces, such as the proof of Frankel conjectures [2], the proof of Hartshorne conjecture [3], and many others [4]- [7].In this note, we want to give a characterization of the complex projective spaces via sections of line bundles.
Results which can be found in [1] [8] and [9] are used freely often without explicit references.Let M be a complex space with a line bundle L. ϑ is the sheaf of germs of sheaf of holomorphic functions, ( ) is the sheaf of germs of holomorphic sections of a line bundle L.

Characterization of the Projective Spaces
In this paper, a characterization of the projective space will be given.
Definition.Let M be a compact complex space with a line bundle L. M is said to be completely intersected with respect to a line bundle L, provided that complex subspace ( ) From the Lemma 1.1 [1] and the proof of theorem 16.2.1 [8], we have Lemma 1.Let V be a compact irreducible complex space.Let F and L be line bundles over V. Let ϕ be an irreducible section of L and put The following sequence of sheaf homomorphisms is exact: where µ is the multiplication by ϕ , ( ) Let M be a n-dimensional compact complex space with a line bundle L. Let 1 , , k ϕ ϕ be linear independent elements of ( ) spect to L, then there is an exact sequence: where ( ) and β is the restriction map.
Proof.The proof is by induction on k.The case k = 0 is trivial.Since M is completely intersected with respect to L, ( ) ( ) are irreducible.Assume the lemma for 1 k − , we have the exact sequence: ∈ from which it follows that 1 , , k ϕ ϕ are linearly dependent, a contradiction.Thus, k ϕ is nontrivial on The exact sequence in Lemma 1 induces the following exact sequence ( ) ( ) ( ) This means that the kernel of the restriction map β is spanned by the restriction of k ϕ to 1 n k V − + .Combin- ing this with the lemma for 1 k − , we obtain the lemma for k.Now we give the main result of this paper.Theorem.Let M be a n-dimensional compact irreducible complex space with a line bundle L. If ( ) = + and M is completely intersected with respect to L, then M is biholomorphic to a complex space n P of dimension n.Proof.Since ( ) is an irreducible divisor by Claim 1, thus , , , ,

( )
0 , H M L is base point free.By Claim 2, n i V − is an irreducible complex subspace of dimension n i − .In particular, 0 V is one point.If  1 Since F is ample, V − is irreducible.By the theorem above, M is biholomorphic to n P .

( ) [ ]
. And due to the definition of n k V − , we have ≥ , a contradiction.It follows that n k