Isosystolic Inequalities for Holomorphic Chains in $\mathbb{C}P^{n}$

TL;DR

This paper introduces the holomorphic $k$-systole for Hermitian metrics on complex projective space and proves the Fubini-Study metric locally minimizes this invariant among Gauduchon metrics, with applications to constructing metrics with specific systolic properties.

Contribution

It defines the holomorphic $k$-systole and proves the Fubini-Study metric's local minimality within Gauduchon metrics, providing new insights into systolic inequalities in complex geometry.

Findings

Fubini-Study metric locally minimizes volume-normalized holomorphic $(n-1)$-systole among Gauduchon metrics.

Construction of Gauduchon metrics on $ ext{CP}^2$ close to Fubini-Study with non-holomorphic systoles.

Introduction of holomorphic $k$-systole as a new geometric invariant.

Abstract

We introduce the \emph{holomorphic $k$-systole} of a Hermitian metric on $\mathbb{C}P^n$, defined as the infimum of areas of homologically non-trivial holomorphic $k$-chains. Our main result establishes that, within the set of Gauduchon metrics, the Fubini-Study metric locally minimizes the volume-normalized holomorphic $(n-1)$-systole. As an application, we construct Gauduchon metrics on $\mathbb{C}P^2$ arbitrarily close to the Fubini-Study metric whose homological $2$-systole is realized by non-holomorphic chains.