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

Luciano L. Junior

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arXiv · 2507.23156

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

Luciano L. Junior

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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.

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Luciano L. Junior. 2025-07-30. Isosystolic Inequalities for Holomorphic Chains in $\mathbb{C}P^{n}$. https://arxiv.org/abs/2507.23156

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