Gromov's inequality for complex projective space

We now discuss systolic inequalities on projective spaces.

Theorem 5.1 (M. Gromov)   Let  ${\mathcal G}$ be a Riemannian metric on complex projective space ${\mathbb{C}\mathbb{P}}^n$. Then

$$\stsys _2({\mathcal G})^n \leq n! \vol _{2n}({\mathcal G});$$

equality holds for the Fubini-Study metric on  ${\mathbb{C}\mathbb{P}}^n$.

Proof. Following Gromov's notation in [Gro99, Theorem 4.36], we let

$$\alpha\in H_2({\mathbb{C}\mathbb{P}}^n;{\mathbb{Z}})={\mathbb{Z}}$$ (5.1)

be the positive generator in homology, and let

$$\omega\in H^{2}({\mathbb{C}\mathbb{P}}^n;{\mathbb{Z}})={\mathbb{Z}}$$

be the dual generator in cohomology. Then the cup power $\omega^n$ is a generator of  $H^{2n}({\mathbb{C}\mathbb{P}}^n;{\mathbb{Z}})={\mathbb{Z}}$. Let  $\eta \in \omega$ be a closed differential 2-form. Since wedge product $\wedge$ in  $\Omega^*(X)$ descends to cup product in $H^*(X)$, we have

$$1= \int_{{\mathbb{C}\mathbb{P}}^{n}} \eta^{\wedge n} .$$ (5.2)

Now let  ${\mathcal G}$ be a metric on  ${\mathbb{C}\mathbb{P}}^n$. Recall that the pointwise comass norm for a simple $k$-form coincides with the natural Euclidean norm on $k$-forms associated with  ${\mathcal G}$. In general, the comass is defined as follows.

Definition 5.2   The comass of an exterior $k$-form is its maximal value on a $k$-tuple of unit vectors.

The comass norm of a differential $k$-form is, by definition, the supremum of the pointwise comass norms. Then by the Wirtinger inequality we obtain

$$\begin{aligned}1 & \leq \int_{{\mathbb{C}\mathbb{P}}^n} \Vert \... ...n \vol _{2n}({\mathbb{C}\mathbb{P}}^n,{\mathcal G}) \end{aligned}$$

where  $\Vert \; \Vert _\infty$ is the comass norm on forms. See [Gro99, Remark 4.37] for a discussion of the constant in the context of the Wirtinger inequality. A more detailed discussion appears in [Ka07, Section 13.1].

The infimum of (5.3) over all  $\eta \in \omega$ gives

$$1 \leq n! \left( \Vert \omega \Vert^* \right)^n \vol _{2n} \left( {\mathbb{C}\mathbb{P}}^n, {\mathcal G}\right),$$ (5.4)

where $\Vert\;\Vert^*$ is the comass norm in cohomology. Denote by $\Vert\;\Vert$ the stable norm in homology. Recall that the normed lattices  $(H_2(M;{\mathbb{Z}}), \Vert\;\Vert)$ and  $(H^2(M;{\mathbb{Z}}), \Vert\;\Vert^*)$ are dual to each other [Fed69]. Therefore the class $\alpha$ of (5.1) satisfies

$$\Vert\alpha \Vert = \frac{1}{\Vert\omega \Vert^*},$$

and hence

$$\stsys _2({\mathcal G})^n = \Vert \alpha \Vert^n \leq n! \vol _{2n}({\mathcal G}) .$$ (5.5)

Equality is attained by the two-point homogeneous Fubini-Study metric, since the standard line  ${\mathbb{C}\mathbb{P}}^1 \subset {\mathbb{C}\mathbb{P}}^n$ is calibrated by the Fubini-Study Kahler $2$-form, which satisfies equality in the Wirtinger inequality at every point. $\qedsymbol$

Example 5.3   Every metric  ${\mathcal G}$ on the complex projective plane satisfies the optimal inequality

$$\stsys _2({\mathbb{C}\mathbb{P}}^2,{\mathcal G})^2 \leq 2 \vol _{4}({\mathbb{C}\mathbb{P}}^2,{\mathcal G}).$$