A family of complex nilmanifolds with infinitely many real homotopy types

TL;DR

This paper constructs a one-parameter family of 8-dimensional nilpotent Lie algebras with complex structures, demonstrating infinitely many distinct real homotopy types of nilmanifolds with such structures, and explores compatible metrics.

Contribution

It introduces a new family of complex nilpotent Lie algebras parameterized by a real variable, showing the existence of infinitely many homotopy types of nilmanifolds with complex structures.

Findings

Infinitely many real homotopy types of 8-dimensional nilmanifolds with complex structures.

Construction of balanced Hermitian and generalized Gauduchon metrics on these nilmanifolds.

Abstract

We find a one-parameter family of non-isomorphic nilpotent Lie algebras $\mathfrak{g}_a$, with $a \in [0,\infty)$, of real dimension eight with (strongly non-nilpotent) complex structures. By restricting $a$ to take rational values, we arrive at the existence of infinitely many real homotopy types of $8$-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.