Gromov's inequality for complex projective space

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality

${\displaystyle \mathrm {stsys} _{2}{}^{n}\leq n!\;\mathrm {vol} _{2n}(\mathbb {CP} ^{n})}$,

valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here ${\displaystyle \operatorname {stsys_{2}} }$ is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line ${\displaystyle \mathbb {CP} ^{1}\subset \mathbb {CP} ^{n}}$ in 2-dimensional homology.

The inequality first appeared in Gromov (1981) as Theorem 4.36.

The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms.