Symplectic real Bott manifolds

A real Bott manifold is the total space of an iterated $\RP ^1$-bundles over a point, where each $\RP^1$-bundle is the projectivization of a Whitney sum of two real line bundles. In this paper, we characterize real Bott manifolds which admit a symplectic form. In particular, it turns out that a real Bott manifold admits a symplectic form if and only if it is cohomologically symplectic. In this case, it admits even a K\"{a}hler structure. We also prove that any symplectic cohomology class of a real Bott manifolds can be represented by a symplectic form. Finally, we study the flux of a symplectic real Bott manifold.


Introduction
A real Bott tower (of height n) is a sequence of RP 1 -bundles: where each RP 1 -bundle M i → M i−1 is the projectivization of a Whitney sum of two real line bundles on M i−1 . Each M i is called a real Bott manifold. Clearly M 1 = RP 1 and M 2 = (RP 1 ) 2 or a Klein bottle. If every bundle in the tower is trivial, then M n = (RP 1 ) n . However, there are many choices of non-trivial bundles at each stage in the tower and it is known that there are many different diffeomorphism classes in real Bott manifolds ( [5], [6]). A real Bott manifold is also an example of a real toric manifold which admits a flat Riemannian metric ( [5]).
Although orientable ones occupy a small portion in all real Bott manifolds ( [3]), the number of orientable ones of dimension n approaches infinity as n approaches infinity. Among those orientable ones, some are symplectic, i.e., admit a symplectic form. In this paper we give a complete characterization of symplectic real Bott manifolds (Theorem 3.1). In particular, we prove that among real Bott manifolds M the following are equivalent: We remark that the implication (3) ⇒ (2) ⇒ (1) always holds but the reverse implications (1) ⇒ (2) and (2) ⇒ (3) do not hold in general as is well-known. For example, C P 2 # C P 2 is cohomologically symplectic but not symplectic because it does not admit an almost complex structure and a certain T 2 -bundle over T 2 constructed in [8] is symplectic but does not admit a Kähler structure. This paper is organized as follows. In Section 2 we recall the quotient description of real Bott manifolds. In Section 3 we state and prove our main theorem. In Section 4 we study the flux group of a symplectic real Bott manifold.
Throughout this paper, all cohomology will be de Rham cohomology over R.

Quotient description of real Bott manifolds
In this section, we recall the quotient description of real Bott manifolds (see [5] and [6] for details) and observe the cohomology ring of a real Bott manifold.
Let B(n) be the the set of n × n upper triangular (0, 1) matrices with zero diagonal entries. For a matrix A ∈ B(n), A i j denotes the (i, j) entry of A and A i (respectively, A j ) denotes the i-th row (respectively, j-th column) of A. Let S 1 be the unit circle in C. For z ∈ S 1 and a ∈ Z /2 = {0, 1}, we set z(a) := a if a = 0 andz if a = 1. We then define the involution a i on T n := (S 1 ) n by . . , z n (A i n )) for i = 1, . . . , n. Let G(A) denote the transformation group on T n generated by a i 's. Then the quotient space M (A) := T n /G(A) is known to be a real Bott manifold and every real Bott manifold can be obtained as M (A) for some A ∈ B(n). Although A is not necessarily uniquely determined by a real Bott manifold, A contains all geometrical information on M (A). For example, (see [5]). It is also helpful to describe M (A) as the quotient of R n by affine transformations. In fact, let Γ(A) denote the affine transformation group on R n generated by s i 's defined by for i = 1, . . . , n. Then, an exponential map from R to S 1 sending u to exp(2π Let du 1 , . . . , du n denote the standard 1-forms on R n . Since each du j is invariant under parallel translations on R n , it descends to a closed 1-form on T n ∼ = R n / Z n , which we also denote by du j . The (de Rham) cohomology ring H * (T n ) of T n is the exterior algebra in n variables [du 1 ], . . . , [du n ] over R, where [du j ] denotes the cohomology class represented by the 1-form du j . It follows from (2.1) or (2.3) that the endomorphism a * i of H * (T n ) induced by a i ∈ G(A) is given by (see [2, Theorem 2.4 in p.120] for example), where the right hand side denotes the G(A)-invariants in H * (T n ).

Main theorem
The following is our main theorem in this paper.
Proof of (1) ⇒ (2). Assume that there exists a de Rham cohomology class α ∈ H 2 (M (A)) such that α n = 0. We identify H * (M (A)) with H * (T n ) G(A) by (2.5). Then it follows from Corollary 2.1 that we can write α uniquely as Thus α n = 0 implies the condition (2). Proof of (2) ⇒ (4). Assume that A ∈ B(2n) satisfies the condition (2), namely A j k = A j k+n for k = 1, . . . , n. Then we identify R 2n with C n by z k := u j k + √ −1u j k+n for k = 1, . . . , n. Consider the standard Hermitian metric on C n . Then, Γ(A) acts on C n as biholomorphisms and isometries. In fact, through the above identification, it follows from (2.3) that the action of s i ∈ Γ(A) on C n is given by Finally, we shall prove the last statement in the theorem. As observed above, α ∈ H 2 (M (A)) is of the form (3.1). We then define the differential closed 2-form ω on R 2n by Comparing (3.1) with (3.2), one sees that the condition α n = 0 implies that ω n is nowhere zero. Thus ω is a symplectic form on R 2n . Since ω is invariant under the Γ(A)-action on R 2n , ω descends to a symplectic form on the quotient M (A) = R 2n /Γ(A) and this represents the given class α.  Table 1] . Finally we note that if then M (A) is orientable by (2.2), but not symplectic. Therefore the class of symplectic real Bott manifolds is strictly smaller than that of orientable real Bott manifolds.

The flux group
In this section, we will study the flux group of a symplectic real Bott manifold. For that, we recall the definition of a flux group for a general symplectic manifold.
Let (M, ω) be a closed symplectic manifold. A diffeomorphism φ : M → M is called a symplectomorphism if φ * ω = ω and the group of symplectomorphisms of (M, ω) is denoted by Symp(M, ω). Associated to a smooth function f : M → R, the Hamiltonian vector field X f is defined by i X f ω = df . For a one-parameter family {f t } 0≤t≤1 of functions, we obtain a one parameter family {X ft } 0≤t≤1 of Hamiltonian vector fields, and integrating {X ft }, we obtain a one-parameter family {φ t } 0≤t≤1 of diffeomorphisms defined by The time-one map φ 1 is a symplectomorphism and called a Hamiltonian diffeomorphism. It is known that all Hamiltonian diffeomorphisms of (M, ω) form a subgroup, denoted Ham(M, ω), of the identity component Symp 0 (M, ω) of Symp(M, ω). For a symplectic isotopy {φ t }, that is, an isotopy through symplectomorphisms, we obtain a one-parameter family {X t } of vector fields define by The flux of {φ t } is then defined to be It is known that the flux depends only on the homotopy class of symplectic isotopies with fixed end points φ 0 = id and φ 1 , so that it defines a homomorphism where Γ ω is the image of the fundamental group π 1 (Symp 0 (M, ω)) by the flux and called the flux group of (M, ω). The solution of the flux conjecture ( [7]) says that the subgroup Γ ω of H 1 (M ) is closed and discrete. According to [1], the kernel of Flux is exactly equal to Ham(M, ω), in other words, we have an exact sequence Now, we consider the flux of a symplectic real Bott manifold. c j,k du j ∧ du k .
Since M (A) = T 2n /G(A) and A p = 0 for p = 1, . . . , 2r, the multiplication of S 1 on the p-th coordinate on T 2n for 1 ≤ p ≤ 2r descends to an S 1 -action on M (A) and defines a symplectic isotopy {φ p t }. The one-parameter family {X p t } of vector fields associated with {φ p t } is then ∂/∂u p (possibly up to a non-zero constant), so that it follows from (4.1) and (4. where q = p + r if 1 ≤ p ≤ r and q = p − r if r + 1 ≤ p ≤ 2r. This shows that Γ ω spans H 1 (M (A)) over R. Since Γ ω is closed and discrete in H 1 (M (A)) as remarked before, it must be a lattice group of H 1 (M (A)) of full rank.