Resumen de Geometry of nilmanifolds with invariant complex structure

Adela Latorre Larrodé

• In this thesis we focus our attention on a special class of compact manifolds, known as nilmanifolds, and study their complex geometry up to (real) dimension eight. Our main objectives are to construct all invariant complex structures on such spaces, analyze several cohomological aspects in complex dimension 3, and investigate new geometric structures and Hermitian metrics which arise in complex dimension 4. We next give a short description of each chapter.

  Chapter 1 is introductory. We review the basic concepts that will be used along this work and fix the notation.

  In Chapter 2 we investigate cohomological properties of 6-dimensional nilmanifolds with invariant complex structure. First, we study the behaviour under holomorphic deformation of some properties related to the d-dbar-lemma condition that can be defined in terms of the Bott-Chern cohomology groups. Then, we focus on the problem of cohomological decomposition, paying particular attention to the real decomposition at the second stage.

  Chapter 3 is devoted to the problem of constructing invariant complex structures on nilmanifolds of arbitrary dimension 2n. More concretely, we provide an strategy to find any complex structure J on any 2n-dimensional nilpotent Lie algebra without the need of knowing the involved algebras in advance. Indeed, two methods are introduced according to the degree of nilpotency of the complex structure J to be constructed. The combination of these two approaches allows to construct invariant complex structures on nilmanifolds of any even dimension. As an application, we recover the classification of complex structures on nilpotent Lie algebras of dimensions four and six. We also start the study of dimension eight, which is completed in Chapter 4. As a consequence, we parametrize every 8-dimensional nilmanifold endowed with an invariant complex structure.

  The previous classification result is used in Chapter 5 with the aim of analyzing some geometric structures that do not appear in dimensions 4 and 6. On the one hand, we concentrate on Hermitian metrics, paying attention to the similarities and differences among SKT, astheno-Kähler, and generalized Gauduchon metrics. On the other hand, we study holomorphic symplectic and pseudo-Kähler structures.