On the Real Homotopy Type of Generalized Complex Nilmanifolds

Adela Latorre,Raquel Villacampa,Luis Ugarte

doi:10.3390/math8091562

Abstract

We prove that for any n≥4, there are infinitely many real homotopy types of 2n-dimensional nilmanifolds admitting generalized complex structures of every type k, for 0≤k≤n.

Highlights

Nilmanifolds constitute a well-known class of compact manifolds providing interesting explicit examples of geometric structures with special properties
We address the problem of finding a family of nilmanifolds with infinitely many real homotopy types that admit complex and symplectic structures, and generalized complex structures of every possible type
Let us start recalling some general results about homotopy theory and minimal models, with special attention to the class of nilmanifolds

Summary

Introduction

Nilmanifolds constitute a well-known class of compact manifolds providing interesting explicit examples of geometric structures with special properties. The existence of infinitely many real homotopy types of 8-dimensional nilmanifolds with a complex structure (having special Hermitian metrics) is proved in [10]. These nilmanifolds do not admit any symplectic form. There are infinitely many real homotopy types of 8-dimensional nilmanifolds admitting generalized complex structures of every type k, for 0 ≤ k ≤ 4. In dimension 6 there are nilmanifolds admitting generalized complex structures of every possible type, their real homotopy types are finite. There are infinitely many complex homotopy types of 2n-dimensional compact non-Kähler manifolds admitting generalized complex structures of every type k, for 0 ≤ k ≤ n

The family of Nilmanifolds Nα

Generalized Complex Structures on the Nilmanifolds Nα

The Nilmanifolds Nα and Their Minimal Model