Contact Moishezon threefolds with second Betti number one

We prove that the only contact Moishezon threefold having second Betti number equal to one is the projective space.


Introduction.
A compact complex manifold X of dimension 2n + 1 is said to be a contact manifold if there is a vector bundle sequence where T X is the tangent bundle of X and F a sub-bundle of rank 2n such that the induced map 2 is everywhere non-degenerate. Properly speaking (X, F ) is a contact manifold; the line bundle L is called the contact line bundle. It is easy to see that −K X = (n + 1)L, that is in case X is a threefold: We refer e.g. to [12] and [2] for details. There are basically two methods to construct compact contact manifolds.
• A simple Lie group gives rise to a Fano contact manifold X by taking the unique closed orbit for the adjoint action of the Lie group on the projectivised Lie algebra; we refer to [1]. Unless the group is of type A, we have b 2 (X) = 1. Specifically, this construction includes P 2n+1 , P(T P n+1 ), Grassmannians of lines on quadrics, and some exceptional homogeneous spaces.
• Given any compact complex manifold M , the projectivised tangent bundle P(T M ) is a contact manifold.
A famous conjecture of LeBrun and Salamon [13] claims that there are no other projective contact manifolds. If b 2 (X) ≥ 2, this is settled by [11] and [5]. For results in case b 2 (X) = 1, we refer to [1][2][3]7,8,12,15]. Since there is no known example of a compact contact manifold not in the above list, one might wonder whether the projectivity assumption in the conjecture of LeBrun and Salamon is really necessary. Dropping the projectivity assumption, it seems reasonable to assume first that X is not too wild, i.e. X is in class C, which is to say that X is bimeromorphic to a compact Kähler manifold.
In [6] it has been shown that a contact threefold in class C which is not rationally connected must be of the form X = P(T M ) with a Kähler surface M. Thus it remains in dimension 3 to treat rationally connected varieties in class C. Notice that these are automatically Moishezon spaces, i.e., they carry three algebraically independent meromorphic functions, see Proposition 2.2. In fact, the rational connectedness of X implies that H 2 (X, O X ) = 0. In this short note we treat the case that b 2 (X) = 1.
In the projective case, this theorem has first been shown by Ye [16].

Preliminaries.
We will make heavily use of the following theorem of Kollár [9] and Nakamura.
Next we collect some basic properties of rationally connected manifolds. Recall that a compact manifold in class C is rationally connected if two general points in X can be joined by a chain of rational curves. For the benefit of the reader, we list the following well-known properties and include indications on the proof. (1) X is simply connected; (2) H q (X, O X ) = 0 for all q ≥ 1; in particular X is Moishezon. Proof.
(1) We refer to [4,Corollary 5.7]. Notice that in [4], the manifold is supposed to be Kähler. Since however X is bimeromorphically equivalent to a Kähler manifold, we may choose a birational holomorphic mapX → X withX Kähler, given by a sequence of blow-ups with smooth centers. Then we apply Campana's theorem onX and use the basic fact π 1 (X) = π 1 (X) (it suffices to check that for a single blow-up along a submanifold).
(2) Since X is rationally connected, there exists a rational curve C ⊂ X such that the tangent bundle T X |C is ample, see [10,IV.3.7] (the proof works for manifolds in class C as well). From this fact it follows easily H 0 (X, Ω q X ) = 0, hence by Hodge duality H q (X, O X ) = 0 for q > 0. We refer to [11,IV.3] for details. In order to show that X is Moishezon, observe that H q (X, OX ) = 0 for positive q, in particular H 2 (X, OX ) = 0. Thus by Kodaira's classical theoremX is projective and therefore X is Moishezon.
(3) Suppose Pic(X) contains a torsion element. Thus there is a non-trivial line bundle M such that M ⊗m O X for some positive number m. As a consequence, there is a finiteétale cover f :X → X such that f * (M ) OX . This contradicts the simply connectedness of X.

Proof of the theorem.
To start the proof of the main theorem, we first observe that X is uniruled, see [6,Theorem 2.2]. Furthermore, X is rationally connected, for otherwise by the main Theorem in [6] X is isomorphic to P(T M ) with a Kähler surface M and b 2 (X) ≥ 2, a contradiction to our assumption. In particular by Proposition 2.2, X is Moishezon, simply connected and PicX Z. Let O X (1) be the effective generator of Pic(X). Since the canonical line bundle K X is divisible by 2 by (1.1), we have with an even integer m. Since X is uniruled, m must be negative, see [9,Theorems (5.3.2)  Since 0 (X being simply connected), and b 2 = b 4 = 1 by our assumption). Hence (3.1) Since the contact form gives an isomorphism 2 F = L, we have c 1 (F ) = L.