Dolbeault formality for complex nilmanifolds
Tommaso Sferruzza, Misha Verbitsky
TL;DR
This paper investigates the formality of the Dolbeault differential graded algebra (DGA) on complex nilmanifolds, establishing that it is formal only in the case of tori and abelian complex structures.
Contribution
It proves that the Dolbeault DGA of a complex nilmanifold is formal exclusively for tori, and the (0,p)-forms algebra is formal only with abelian complex structures.
Findings
Dolbeault DGA is formal only for tori.
(0,p)-forms algebra is formal only with abelian structures.
Formality characterizes specific geometric structures.
Abstract
A quasi-isomorphism of differential graded algebras (DGA) is a multiplicative map inducing an isomorphism on cohomology. A DGA is called formal if it can be connected by a chain of quasi-isomorphisms to its cohomology algebra. We prove that the Dolbeault DGA of a complex nilmanifold is formal only if it is a torus, and the Dolbeault algebra of (0,p)-forms is formal if and only if the complex structure is abelian.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra