Symplectic Pairs and Intrinsically Harmonic Forms ^†

Abstract

In this short note, we prove two properties of symplectic pairs on a four-manifold: firstly we prove that two transversal orientable foliations of codimension two, which are taut for the same Riemannian metric, are the characteristic foliations of a symplectic pair; secondly, we characterize intrinsically harmonic 2-forms of rank two as part of a symplectic pair.

1. Introduction

A symplectic pair on a smooth manifold [1,2] is a pair of non-trivial closed 2-forms $(\omega ,\eta )$ of constant and complementary ranks, for which $\omega$ restricts to a symplectic form on the leaves of the kernel foliation of $\eta$, and vice versa.

On a four-manifold M, a symplectic pair $(\omega ,\eta )$ can be equivalently defined by a pair of symplectic forms $({\mathsf{\Omega }}_{+},{\mathsf{\Omega }}_{-})$ satisfying the following conditions:

$${\mathsf{\Omega }}_{+}^2=-{\mathsf{\Omega }}_{-}^2,{\mathsf{\Omega }}_{+}\wedge {\mathsf{\Omega }}_{-}=0.$$

(1)

In this case, the forms $\omega$ and $\eta$ are given by $\omega =\frac{1}{2}({\mathsf{\Omega }}_{+}+{\mathsf{\Omega }}_{-})$ and $\eta =\frac{1}{2}({\mathsf{\Omega }}_{+}-{\mathsf{\Omega }}_{-})$.

Several interesting examples and constructions are given in [1], especially on closed four-manifolds. It is also observed that it is possible to construct compatible metrics, making both the characteristic foliations taut (i.e., with minimal leaves).

Recall that a differential form on a manifold is called intrinsically harmonic [3] if it is harmonic with respect to some Riemannian metric. With respect to a compatible metric, each 2-form of the pair forming the symplectic pair is harmonic.

The aim of this paper is two-fold: firstly, we prove that on a four-dimensional orientable manifold, two complementary orientable foliations of dimension 2 which are taut for some metric are, in fact, the characteristic foliations of a symplectic pair; secondly, we show that, on a closed four-dimensional manifold, an intrinsically harmonic 2-form of rank 2 is necessarily one of the 2-forms composing a symplectic pair.

All the objects considered in this paper are assumed to be $C^{\infty }$.

2. Preliminaries on Symplectic Pairs

In this section, we recall the main objects studied in this article and some basic results needed in the next sections.

Definition 1 

([1,2]). Let M be a $2n$-dimensional manifold. A pair of closed 2-forms $(\omega ,\eta )$ is called a symplectic pair of type $(k,n-k)$ if they have constant ranks $2k$ and $2(n-k)$, respectively, and, moreover, ${\omega }^{2k}\wedge {\eta }^{2(n-k)}$ is a volume form.

A symplectic pair gives rise to two symplectic forms:

$${\mathsf{\Omega }}_{+}=\omega +\eta ,{\mathsf{\Omega }}_{-}=\omega -\eta$$

on M and on ${(-1)}^{n-p}M$, respectively, where $-M$ denotes the oriented manifold obtained by reversing the orientation of M. To make the definition interesting, we assume that $k>0$ and $n>k$. Then, when M has dimension four—the case of interest in the present article—a symplectic pair on M can only be of type $(1,1)$ and, in particular, M is symplectic for both orientations.

In dimension four, a symplectic pair $(\omega ,\eta )$ can be equivalently defined by a pair of symplectic forms $({\mathsf{\Omega }}_{+},{\mathsf{\Omega }}_{-})$ satisfying

$${\mathsf{\Omega }}_{+}^2=-{\mathsf{\Omega }}_{-}^2,{\mathsf{\Omega }}_{+}\wedge {\mathsf{\Omega }}_{-}=0.$$

The symplectic pair is then given by

$$\left(\frac{{\mathsf{\Omega }}_{+}+{\mathsf{\Omega }}_{-}}{2},\frac{{\mathsf{\Omega }}_{+}-{\mathsf{\Omega }}_{-}}{2}\right),$$

and we say that $({\mathsf{\Omega }}_{+},{\mathsf{\Omega }}_{-})$ arises from a symplectic pair.

The kernels of $\omega$ and $\eta$ are integrable complementary distributions and therefore integrate to a pair of transverse foliations ${\mathcal{F}}_{\omega }$ and ${\mathcal{F}}_{\eta }$ called characteristic foliations [1] such that

$$T{\mathcal{F}}_{\omega }=ker\omega \mathrm{and}T{\mathcal{F}}_{\eta }=ker\eta .$$

Each form is symplectic on the leaves of the foliation induced by the other form and, moreover, ${\mathcal{F}}_{\omega }$ and ${\mathcal{F}}_{\eta }$ are symplectically orthogonal with respect to both the symplectic forms ${\mathsf{\Omega }}_{+}$ and ${\mathsf{\Omega }}_{-}$.

By Rummler and Sullivan’s criterion (see [4]), the characteristic foliation of a closed 2-form is taut, which means that there exists a Riemannian metric for which the leaves are minimal. For a symplectic pair, it is possible to construct a Riemannian metric, making the foliations orthogonal and both with minimal leaves (see [1] and the discussion after Definition 5.6 in [5]).

3. Taut Foliations and Symplectic Pairs

It is shown in [6] that on a four-dimensional orientable manifold “two taut make one symplectic”, which means that the existence of two complementary orientable 2-dimensional taut foliations implies that the manifold is symplectic.

In this section, we see that “two taut make a symplectic pair”, proving that the two foliations are, in fact, the characteristic foliations of a symplectic pair as shown in the following result:

Theorem 1. 

Let M be an orientable four-dimensional manifold endowed with two transverse and complementary orientable foliations $\mathcal{F}$ and $\mathcal{G}$ of dimension 2. If $\mathcal{F}$ and $\mathcal{G}$ are orthogonal and have minimal leaves for some Riemannian metric on M, then they are the characteristic foliations of a symplectic pair.

Proof. 

Let g be a metric for which $\mathcal{F}$ and $\mathcal{G}$ are orthogonal and have minimal leaves. Consider g-orthogonal almost complex structures $J_1,J_2$, respectively, on $T\mathcal{F}$ and $T\mathcal{G}$. Then, we have two almost complex structures on $TM$ given by

$$J_{\pm }=J_1\oplus (\pm J_2).$$

Let ${\mathsf{\Omega }}_{\pm }(X,Y)=g(X,J_{\pm }Y)$. For ∇, the Levi–Civita connection of g and $X,Z$ vector fields on M, we have (see Appendix in [6] for a proof):

$$d{\mathsf{\Omega }}_{\pm }(X,J_{\pm }X,Z)=g([X,J_{\pm }X],J_{\pm }Z)-g({\nabla }_{X}X+{\nabla }_{J_{\pm }X}{J}_{\pm }X,Z).$$

To prove that $d{\mathsf{\Omega }}_{\pm }=0$, it is enough to prove that $d{\mathsf{\Omega }}_{\pm }$ vanishes when calculated on any triple of linearly independent vector fields. We can choose a local basis such that $X,J_{\pm }X$ are tangent to $\mathcal{F}$ and $Z,J_{\pm }Z$ are tangent to $\mathcal{G}$. Any triple of vectors fields of the local basis has the form $(U,J_{\pm }U,V)$ for some $U,V$ in the local basis.

By the minimality of the leaves, we have $g({\nabla }_{X}X+{\nabla }_{J_{\pm }X}{J}_{\pm }X,Z)=0$ (see [7] for example). Frobenius’ theorem and the orthogonality of $\mathcal{F}$ and $\mathcal{G}$ imply $g([X,J_{\pm }X],J_{\pm }Z)=0$. This implies that the 2-forms ${\mathsf{\Omega }}_{\pm }$ are closed and therefore symplectic.

Since ${\mathsf{\Omega }}_{\pm }$ are symplectic, there is a unique isomorphism A of the tangent bundle of M, called recursion operator, such that

$${\mathsf{\Omega }}_{+}(X,Y)={\mathsf{\Omega }}_{-}(AX,Y).$$

In fact, A is the composition of the usual musical isomorphisms.

In our case, the recursion operator A is the identity on one foliation and minus the identity on the other one. In particular, A is not the identity itself, but its square is the identity. Thus, by Theorem 3 in [8] the pair $({\mathsf{\Omega }}_{+},{\mathsf{\Omega }}_{-})$ arises from a symplectic pair. □

4. Intrinsically Harmonic 2-Forms

A differential form $\omega$ on an n-dimensional manifold is called intrinsically harmonic [3] if it is harmonic with respect to some Riemannian metric. An intrinsically harmonic form is a fortiori closed, and then the main problem is to give necessary and sufficient conditions under which a closed form is intrinsically harmonic.

In fact, only the forms of degrees 1 and $n-1$ are quite well understood. A classical theorem of Calabi [3] answers the question for 1-forms with non-degenerate zeros, and Honda [9] proves the dual case of $(n-1)$-form.

In 2007, Volkov [10] was able to drop the condition on the zeros of the 1-form, giving a complete characterization.

In general, the forms of degrees strictly between 1 and $n-1$ present additional problems. One of this difficulties is illustrated in [10], where the author gives an example of a closed 2-form of rank 2 on a 4-dimensional manifold, which is not intrinsically harmonic.

Observe that any symplectic form is harmonic with respect to a compatible metric because its Hodge dual is, up to a constant, a power of the symplectic form.

On a 4-dimensional manifold, a non-vanishing 2-form of constant rank has either rank 2 or 4. Vanishing forms and symplectic forms (of rank 4) are intrinsically harmonic, so let us consider the 2-forms of rank 2. We start with the following result of linear algebra.

Lemma 1. 

Let ω be a 2-form of rank 2 on a four-dimensional Euclidean vector space $(W,g)$ and let 🟉 be the Hodge operator with respect to g. Then $🟉\omega$ also has rank 2.

Proof. 

Since the rank of $\omega$ is 2, then it has 2-dimensional kernel. Let $W\subset W$ be the kernel of $\omega$ and let $\{e_1,e_2\}$ be an othonormal basis of V. Complete $\{e_1,e_2\}$ to an orthonormal basis $\{e_1,e_2,e_3,e_4\}$ of W. Then $\omega$ is (up to a non-zero constant) equal to $e_3\wedge e_4$ and $🟉\omega$ is equal (up to a non-zero constant) to $e_1\wedge e_2$. □

We can now give the proof of the main theorem:

Theorem 2. 

A closed 2-form of constant rank 2 on an orientable closed four-dimensional manifold M is intrinsically harmonic if and only if it is part of a symplectic pair.

Proof. 

We already observed that on a manifold endowed with a symplectic pair, there exist compatible metrics. With respect to these metrics, each form of the symplectic pair is then closed and co-closed.

On the other end, let $\omega$ be a 2-form of rank 2 which is intrinsically harmonic for some metric g on M. Then $🟉\omega$ has also rank 2 at each point by Lemma 1. Moreover, for $Vo{l}_g$, the volume form associated with g, we have

$$\omega \wedge 🟉\omega ={\left|\right|\omega \left|\right|}^{2}Vo{l}_g$$

which vanishes in a point p only if ${\omega }_p$ does. Then $(\omega ,🟉\omega )$ is a symplectic pair on M. □

5. Conclusions

Theorem 2 is somehow suggested at the end of the last section of [10], but symplectic pairs are not mentioned there. Moreover, the author seems to relegate this possible link to a mere tautological definition. We think instead, that this point of view could reveal some interesting aspects.

Let us consider, for example, the case of $\mathbb{C}{P}^2$. Since it is symplectic, it admits intrinsically harmonic 2-forms of constant rank 4. The existence of a symplectic pair on a manifold implies that its second Betti number $b_2$ satisfies $b_2\ge 2$ and, therefore, no intrinsically harmonic 2-form of constant rank 2 exists.

A more subtle example is given by $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ (the non-trivial $\mathbb{C}{P}^1$-bundle over $\mathbb{C}{P}^1$), which fulfills all the basic topological obstructions to the existence of a symplectic pair. In Example 1 of [10], the author considers, on $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$, the pullback to the total space of a volume form on the base (which has constant rank 2) and proves that it is not intrinsically harmonic. But we can say much more, because, by the results in [11], $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ admits no symplectic pair at all and thus, by Theorem 2, we have the following.

Corollary 1. 

$\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ admits no intrinsically harmonic 2-form of constant rank 2.

Recall that $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ admits a symplectic form and hence the only intrinsically harmonic 2-forms of constant rank, can have rank 4 or 0.

We thus have the following natural question.

Question 1. 

Does $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ admit an intrinsically harmonic 2-form of non-constant rank, which is not symplectic and has at least rank 2 in a point?

The answer is positive, and an example can be constructed as follows.

Example 1. 

Let ω be the pullback to $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ of the Fubini–Study volume form on $\mathbb{C}{P}^1$, and fix any Riemannian metric g on $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$. By the Hodge theorem, the cohomology class of ω has a unique harmonic representative, let us say $\omega +d\alpha$ for some 1-form α. Since ω is not exact (because it is one of the generators of the second cohomology group of $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$), there is a point q, where $\omega +d\alpha$ is non-zero. Then $\omega +d\alpha$ is non-trivial and, in particular, its rank r at q is $r_q\ge 2$. On the other hand, because ${\omega }^2=0$, we have ${(\omega +d\alpha )}^2=2\omega \wedge d\alpha +d{\alpha }^2$, which is exact and, thus, it cannot be a volume form by Stokes’ theorem. This means that there is a point p, where ${(\omega +d\alpha )}^2=0$ and then the rank of ${(\omega +d\alpha )}_p$ is $r_p\le 2$. Therefore, $\omega +d\alpha$ is non-trivial, g-harmonic, cannot have constant rank 2 and cannot be symplectic.

One can try to seek more restricted ranks and ask the following questions:

Question 2. 

Does $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ admit an intrinsically harmonic 2-form of non-constant rank $r\le 4$, which has rank 4 at least in a point?

Question 3. 

Does $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ admit an intrinsically harmonic 2-form of non-constant rank r such that $2\le r\le 4$ and there are points where $r=2$ and $r=4$?

Of course, $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ can be replaced by the blow-up of $\mathbb{C}{P}^2$ in the k point or, more generally, by a closed orientable symplectic 4-manifold, which is non-minimal.