Strong formality below middle degree implies strong formality

TL;DR

The paper proves that strong formality properties are preserved under certain geometric operations and applies these results to specific classes of complex manifolds, extending the understanding of formality in complex geometry.

Contribution

It introduces the notion of $s$-strong formality in the pluripotential setting and characterizes strong formality for complex manifolds based on their dimension and cohomology.

Findings

Hyperplane sections inherit strong formality.

Generalized complete intersections with positive line bundles are strongly formal.

Compact Kähler and $ar{ ext{d}}ar{ ext{d}}$-manifolds with specific cohomology properties are strongly formal.

Abstract

We show that hyperplane sections of strongly formal manifolds inherit strong formality. In particular, this property holds for generalized complete intersections defined by positive line bundles with trivial first de Rham cohomology group. Furthermore, we establish the strong formality of compact K\"ahler manifolds with central cohomology of width $\frac{n}{2}-1$ and, more generally, of compact $\partial\bar{\partial}$-manifolds with no non-trivial multiplicative relations in cohomology below degree $n+2$. These results arise from the notion of $s$-strong formality, which we adapt from a work of Fernandez and Mu\~noz to the pluripotential setting. Specifically, we prove that a compact, connected complex manifold of dimension $n$ with trivial first de Rham cohomology group is strongly formal if and only if it is $(n-1)$-strongly formal.