Sharp systolic inequalities for K\"ahler manifolds

TL;DR

This paper proves sharp inequalities for systolic invariants of K"ahler manifolds, characterizing equality cases and providing bounds for various geometric quantities, with implications for scalar curvature and Gromov's conjecture.

Contribution

It establishes optimal systolic inequalities for K"ahler manifolds and characterizes equality cases, advancing understanding of scalar curvature and geometric invariants.

Findings

Equality cases for complex projective space with Fubini--Study metric.

Refinements for Fano manifolds distinguishing specific varieties.

Bounds for Gromov width, volume, and higher systoles of K"ahler manifolds.

Abstract

We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the $2$-systole and the spherical $2$-systole of compact K\"ahler manifolds, and the stable $2$-systole of Riemannian metrics on a general class of $\mathrm{spin}^c$ manifolds and their products. These bounds attain equality precisely for complex projective space $\mathbb{CP}^n$ equipped with the Fubini--Study metric, and admit further refinements for Fano manifolds which distinguish the complex quadric, cubic, and quartic with their canonical K\"ahler--Einstein structures. We also obtain an algebraic characterization of manifolds admitting K\"ahler metrics with non-negative total scalar curvature, which implies Gromov's rational-essentialness conjecture for K\"ahler metrics. Finally, we prove uniform bounds for the stable $2$-systole of $\mathrm{spin}^c$ manifolds under a general essentialness condition, as well as for the Gromov width, volume, and higher stable systoles of K\"ahler manifolds.