Strong formality of toric and homogeneous compact Kähler manifolds

TL;DR

The paper investigates strong formality beyond ordinary rational formality for compact complex manifolds carrying a rich interaction with complex structure, showing that complete smooth complex toric varieties and compact Kähler homogeneous manifolds are strongly formal over $\mathbb{Q}$ via pluripotential and equivariant methods.A refined notion of strong formality over $\mathbb{Q}$ is developed, framed as triples $(A_{\mathbb{Q}},A_{\mathbb{C}},\varphi)$, and the authors provide criteria and lifting arguments to realize strong formality in these settings.The toric and homogeneous cases are treated through Cartan models for Borel fibrations and intrinsic formality results for complete intersection-type cohomology, yielding explicit constructions that avoid reliance on projectivity.The paper also exhibits a counterexample: a family of cbba’s $A_{\lambda}$ with strong formality over $\mathbb{C}$ but not over $\mathbb{Q}$ for irrational $\lambda$, with the obstruction governed by a nontrivial extension in rational Mixed Hodge structures.Overall, the work delineates the boundary between rational and strong formality in complex geometry and highlights how symmetry and Hodge-theoretic data govern the possibility of lifting formality across coefficient fields.

Abstract

All compact Kähler, or even $\partial\bar\partial$-manifolds, are rationally formal. Not all of them are strongly formal. Yet some of them are: For complete smooth complex toric varieties and homogeneous compact Kähler manifolds we show the stronger property that they are both rationally and strongly formal in a compatible way.

Theorems & Definitions (54)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.3.1
• Remark 2.3.2
• Example 2.3.3
• Definition 2.3.4
• Definition 2.3.5
• Definition 2.3.6
• Example 2.3.7